We give an efficient 0.8395-approximation algorithm for the EPR Hamiltonian. Our improvement comes from a new nonlinear monogamy-of-entanglement bound on star graphs and a refined parameterization of a shallow quantum circuit from previous works. We also prove limitations showing that current methods cannot achieve substantially better approximation ratios, indicating that further progress will require fundamentally new techniques.

We study a variant of QMA where quantum proofs have no relative phase (i.e. non-negative amplitudes, up to a global phase). If only completeness is modified, this class is equal to QMA [arXiv:1410.2882]; but if both completeness and soundness are modified, the class (named QMA+ by Jeronimo and Wu) can be much more powerful. We show that QMA+ with some constant gap is equal to NEXP, yet QMA+ with some *other* constant gap is equal to QMA. One interpretation is that Merlin's ability to "deceive" originates from relative phase at least as much as from entanglement, since QMA(2) $\subseteq$ NEXP.

We study random constraint satisfaction problems (CSPs) in the unsatisfiable regime. We relate the structure of near-optimal solutions for any Max-CSP to that for an associated spin glass on the hypercube, using the Guerra-Toninelli interpolation from statistical physics. The noise stability polynomial of the CSP's predicate is, up to a constant, the mixture polynomial of the associated spin glass. We prove two main consequences: 1) We relate the maximum fraction of constraints that can be satisfied in a random Max-CSP to the ground state energy density of the corresponding spin glass. Since the latter value can be computed with the Parisi formula, we provide numerical values for some popular CSPs. 2) We prove that a Max-CSP possesses generalized versions of the overlap gap property if and only if the same holds for the corresponding spin glass. We transfer results from Huang et al. [arXiv:2110.07847, 2021] to obstruct algorithms with overlap concentration on a large class of Max-CSPs. This immediately includes local classical and local quantum algorithms.

We study the problem of fairly allocating a multiset M of m indivisible items among n agents with additive valuations. Specifically, we introduce a parameter t for the number of distinct types of items and study fair allocations of multisets that contain only items of these t types, under two standard notions of fairness: 1. Envy-freeness (EF): For arbitrary n, t, we show that a complete EF allocation exists when at least one agent has a unique valuation and the number of items of each type exceeds a particular finite threshold. We give explicit upper and lower bounds on this threshold in some special cases. 2. Envy-freeness up to any good (EFX): For arbitrary n, m, and for t ≤ 2, we show that a complete EFX allocation always exists. We give two different proofs of this result. One proof is constructive and runs in polynomial time; the other is geometrically inspired.

We study the performance of local quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) for the maximum cut problem, and their relationship to that of classical algorithms. (1) We prove that every (quantum or classical) one-local algorithm achieves on $D$-regular graphs of girth $> 5$ a maximum cut of at most $1/2 + C/\sqrt{D}$ for $C=1/\sqrt{2} \approx 0.7071$. This is the first such result showing that one-local algorithms achieve a value bounded away from the true optimum for random graphs, which is $1/2 + P_*/\sqrt{D} + o(1/\sqrt{D})$ for $P_* \approx 0.7632$. (2) We show that there is a classical $k$-local algorithm that achieves a value of $1/2 + C/\sqrt{D} - O(1/\sqrt{k})$ for $D$-regular graphs of girth $> 2k+1$, where $C = 2/\pi \approx 0.6366$. This is an algorithmic version of the existential bound of Lyons and is related to the algorithm of Aizenman, Lebowitz, and Ruelle (ALR) for the Sherrington-Kirkpatrick model. This bound is better than that achieved by the one-local and two-local versions of QAOA on high-girth graphs. (3) Through computational experiments, we give evidence that the ALR algorithm achieves better performance than constant-locality QAOA for random $D$-regular graphs, as well as other natural instances, including graphs that do have short cycles. Our experimental work suggests that it could be possible to extend beyond our theoretical constraints. This points at the tantalizing possibility that $O(1)$-local quantum maximum-cut algorithms might be *pointwise dominated* by polynomial-time classical algorithms, in the sense that there is a classical algorithm outputting cuts of equal or better quality *on every possible instance*. This is in contrast to the evidence that polynomial-time algorithms cannot simulate the probability distributions induced by local quantum algorithms.