The edge list model is arguably the simplest input model for graphs, where the graph is specified by a list of its edges. In this model, we study the quantum query complexity of three variants of the triangle finding problem. The first asks whether there exists a triangle containing a target edge and raises general questions about the hiding of a problem's input among irrelevant data. The second asks whether there exists a triangle containing a target vertex and raises general questions about the shuffling of a problem's input. The third asks whether there exists a triangle; this problem bridges the $3$-distinctness and $3$-sum problems, which have been extensively studied by both cryptographers and complexity theorists. We provide tight or nearly tight results for these problems as well as some first answers to the general questions they raise. Furthermore, given any graph with low maximum degree, such as a typical random sparse graph, we prove that the quantum query complexity of finding a length-$k$ cycle in its length-$m$ edge list is $m^{3/4-1/(2^{k+2}-4)\pm o(1)}$, which matches the best-known upper bound for the quantum query complexity of $k$-distinctness on length-$m$ inputs up to an $m^{o(1)}$ factor. We prove the lower bound by developing new techniques within Zhandry's recording query framework [CRYPTO '19] as generalized by Hamoudi and Magniez [ToCT '23]. These techniques extend the framework to treat any non-product distribution that results from conditioning a product distribution on the absence of rare events. We prove the upper bound by adapting Belovs's learning graph algorithm for $k$-distinctness [FOCS '12]. Finally, assuming a plausible conjecture concerning only cycle finding, we show that the lower bound can be lifted to an essentially tight lower bound on the quantum query complexity of $k$-distinctness, which is a long-standing open question.

We consider lifting theorems that transfer lower bounds for two-party communication problems to multiparty communication problems. In particular, following the deterministic Number-on-Forehead (NOF) lifting framework of Yang and Zhang, we study randomized and quantum one-way NOF lifting for composed problems F(z,𝐱) = f(z,G(𝐱)). We work in a one-way NOF model in which only the last player’s view is restricted. The other players have their usual NOF views and communicate as usual, but the last player sees only the gadget output G(𝐱), not the gadget input 𝐱. This kind of restricted-view has appeared in the NOF literature under the name conservative model. Our main contribution is a pair of lifting theorems for this model. In the randomized setting, we show that lifting follows when each preimage G^{-1}(v), the set of gadget inputs with output v, looks pseudorandom to large cylinder intersections. In the quantum setting, we prove an analogous theorem. Equivalently, conservative protocols for F(z,𝐱) = f(z,G(𝐱)) can be converted into two-party one-way protocols for f(z,v) with comparable cost, and the error loss controlled by the corresponding pseudorandomness parameters. Thus, this restriction isolates a setting in which both randomized and quantum one-way NOF lifting can be proved by a direct simulation argument. We prove the required pseudorandomness properties for the generalized inner product gadget over finite fields, using the multiparty character-sum bounds of Yang and Zhang, and for random gadgets, which give non-explicit lifting. As applications, Boolean Hidden Matching yields a randomized-versus-quantum separation in the conservative NOF model, and lifting INDEX gives randomized and quantum conservative NOF lower bounds for O(log n) players.

Quantum entanglement is a key enabling ingredient in diverse applications. However, the presence of unwanted adversarial entanglement also poses challenges in many applications. In this paper, we explore methods to "break" quantum entanglement. Specifically, we construct a dimension-independent k-partite disentangler (like) channel from bipartite unentangled input. We show: For every $d,\ell\ge k$, there is an efficient channel $Λ: \mathbb{C}^{d\ell} \otimes \mathbb{C}^{d\ell} \to \mathbb{C}^{dk}$ such that for every bipartite separable state $ρ_1\otimes ρ_2$, the output $Λ(ρ_1\otimesρ_2)$ is close to a k-partite separable state. Concretely, for some distribution $μ$ on states from $\mathbb{C}^d$, $$ \left\|Λ(ρ_1 \otimes ρ_2) - \int | ψ\rangle \langle ψ|^{\otimes k} dμ(ψ)\right\|_1 \le \tilde O \left(\left(\frac{k^{3}}{\ell}\right)^{1/4}\right). $$ Moreover, $Λ(| ψ\rangle \langle ψ|^{\otimes \ell}\otimes | ψ\rangle \langle ψ|^{\otimes \ell}) = | ψ\rangle \langle ψ|^{\otimes k}$. Without the bipartite unentanglement assumption, the above bound is conjectured to be impossible. Leveraging our disentanglers, we show that unentangled quantum proofs of almost general real amplitudes capture NEXP, greatly relaxing the nonnegative amplitudes assumption in the recent work of QMA^+(2)=NEXP. Specifically, our findings show that to capture NEXP, it suffices to have unentangled proofs of the form $| ψ\rangle = \sqrt{a} | ψ_+ \rangle + \sqrt{1-a} | ψ_- \rangle$ where $| ψ_+ \rangle$ has non-negative amplitudes, $| ψ_- \rangle$ only has negative amplitudes and $| a-(1-a) | \ge 1/poly(n)$ with $a \in [0,1]$. Additionally, we present a protocol achieving an almost largest possible gap before obtaining QMA^R(k)=NEXP$, namely, a 1/poly(n) additive improvement to the gap results in this equality.

We study the probability of Boolean functions with small max influence to become constant under random restrictions. Let f be a Boolean function such that the variance of f is Ω(1) and all its individual influences are bounded by τ. We show that when restricting all but a ρ=Ω((log1/τ)−1) fraction of the coordinates, the restricted function remains nonconstant with overwhelming probability. This bound is essentially optimal, as witnessed by the tribes function =n/Clogn∘Clogn.

Quantum entanglement is a fundamental property of quantum mechanics and it serves as a basic resource in quantum computation and information. Despite its importance, the power and limitations of quantum entanglement are far from being fully understood. Here, we study entanglement via the lens of computational complexity. This is done by studying quantum generalizations of the class NP with multiple unentangled quantum proofs, the so-called QMA(2) and its variants. The complexity of QMA(2) is known to be closely connected to a variety of problems such as deciding if a state is entangled and several classical optimization problems. However, determining the complexity of QMA(2) is a longstanding open problem, and only the trivial complexity bounds ⊆ (2) ⊆ are known.

Abstract. We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order [Formula: see text] sum to at most [Formula: see text], where [Formula: see text] is the number of variables, [Formula: see text] is the tree depth, and [Formula: see text] is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal [ Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. The bounds prior to our work degraded rapidly with [Formula: see text], becoming trivial already at [Formula: see text]. As an application, we obtain, for every integer [Formula: see text], a partial Boolean function on [Formula: see text] bits that has bounded-error quantum query complexity at most [Formula: see text] and randomized query complexity [Formula: see text]. This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis [ SIAM J. Comput., 47 (2018), pp. 982–1038] and Bravyi et al. [ Classical Algorithms for Forrelation, arXiv preprint, 2021]. Prior to our work, the best known separation was polynomially weaker: [Formula: see text] versus [Formula: see text] for any [Formula: see text] [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. As another application, we obtain an essentially optimal separation of [Formula: see text] versus [Formula: see text] for bounded-error quantum versus randomized communication complexity for any [Formula: see text]. The best previous separation was polynomially weaker: [Formula: see text] versus [Formula: see text] (this is implicit in [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]).

We prove that for every decision tree, the absolute values of the Fourier coefficients of given order t≥1 sum to at most (cd/t)t/2(1+logn)(t−1)/2, where n is the number of variables, d is the tree depth, and c>0 is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal (arxiv 2019; FOCS 2020). The bounds prior to our work degraded rapidly with t, becoming trivial already at t=√d.

The threshold degree of a Boolean function f∶{0,1}n→{0,1} is the minimum degree of a real polynomial p that represents f in sign: sgn p(x)=(−1)f(x). A related notion is sign-rank, defined for a Boolean matrix F=[Fij] as the minimum rank of a real matrix M with sgn Mij=(−1)Fij. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits (AC0) is a well-known and extensively studied open problem, with complexity-theoretic and algorithmic applications.

The threshold degree of a Boolean function $f\colon{{\{0,1\}}^n}\to{\{0,1\}}$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: ${sgn}\,p(x)=(-1)^{f(x)}.$ A related notion is sign-rank, defined for a Boolean matrix $F=[F_{ij}]$ as the minimum rank of a real matrix $M$ with ${sgn}\,M_{ij}=(-1)^{F_{ij}}$. Determining the maximum threshold degree and sign-rank achievable by constant-depth circuits (${AC}^{0}$) is a well-known and extensively studied open problem with complexity-theoretic and algorithmic applications. We give an essentially optimal solution to this problem. For any $\epsilon>0,$ we construct an ${AC}^{0}$ circuit in $n$ variables that has threshold degree $\Omega(n^{1-\epsilon})$ and sign-rank $\exp(\Omega(n^{1-\epsilon})),$ improving on the previous best lower bounds of $\Omega(\sqrt{n})$ and $\exp(\tilde{\Omega}(\sqrt{n}))$, respectively. Our results subsume all previous lower bounds on the threshold degree and sign-rank of ${AC}^{0}$ circuits of any given depth, with a strict improvement starting at depth 4. As a corollary, we also obtain near-optimal bounds on the discrepancy, threshold weight, and threshold density of ${AC}^{0}$, strictly subsuming previous work on these quantities. Our work gives some of the strongest lower bounds to date on the communication complexity of ${AC}^{0}$.

Interactive coding, pioneered by Schulman (FOCS '92, STOC '93), is concerned with making communication protocols resilient to adversarial noise. The canonical model allows the adversary to alter a small constant fraction of symbols, chosen at the adversary's discretion, as they pass through the communication channel. Braverman et al. proposed a far-reaching generalization of this model, whereby the adversary can additionally manipulate the channel by removing and inserting symbols. For any ϵ > 0, they showed how to faithfully simulate any protocol in this model with corruption rate up to 1/18 - ϵ, using a constant-size alphabet and a constant-factor overhead in communication. We give an optimal simulation of any protocol in this generalized model of substitutions, insertions, and deletions, tolerating a corruption rate up to 1/4 - ϵ, while keeping the alphabet to a constant size and the communication overhead to a constant factor. This resolves a question due to Gelles (2015). Our corruption tolerance matches an impossibility result for corruption rate 1/4 which holds even for substitutions alone (Braverman and Rao, STOC '11).

Interactive coding, pioneered by Schulman (FOCS 92, STOC 93), is concerned with making communication protocols resilient to adversarial noise. The canonical model allows the adversary to alter a small constant fraction of symbols, chosen at the adversarys discretion, as they pass through the communication channel. Braverman, Gelles, Mao, and Ostrovsky (2015) proposed a far-reaching generalization of this model, whereby the adversary can additionally manipulate the channel by removing and inserting symbols. They showed how to faithfully simulate any protocol in this model with corruption rate up to 1/18, using a constant-size alphabet and a constant-factor overhead in communication. We give an optimal simulation of any protocol in this generalized model of substitutions, insertions, and deletions, tolerating a corruption rate up to 1/4 while keeping the alphabet to a constant size and the communication overhead to a constant factor. Our corruption tolerance matches an impossibility result for corruption rate 1/4 which holds even for substitutions alone (Braverman and Rao, STOC 11).