We connect three distinct lines of research that have recently explored extensions of the classical LOCAL model of distributed computing: A. distributed quantum computing and non-signaling distributions [e.g. STOC 2024], B. finitely-dependent processes [e.g. Forum Math. Pi 2016], and C. locality in online graph algorithms and dynamic graph algorithms [e.g. ICALP 2023]. We prove new results on the capabilities and limitations of all of these models of computing, for locally checkable labeling problems (LCLs). We show that all these settings can be sandwiched between the classical LOCAL model and what we call the randomized online-LOCAL model. Our work implies limitations on the quantum advantage in the distributed setting, and we also exhibit a new barrier for proving tighter bounds. Our main technical results are these: 1. All LCL problems solvable with locality $O(\log^\star n)$ in the classical deterministic LOCAL model admit a finitely-dependent distribution with locality $O(1)$. This answers an open question by Holroyd [2024], and also presents a new barrier for proving bounds on distributed quantum advantage using causality-based arguments. 2. In rooted trees, if we can solve an LCL problem with locality $o(\log \log \log n)$ in the randomized online-LOCAL model (or any of the weaker models, such as quantum-LOCAL), we can solve it with locality $O(\log^\star n)$ in the classical deterministic LOCAL model. One of many implications is that in rooted trees, $O(\log^\star n)$ locality in quantum-LOCAL is not stronger than $O(\log^\star n)$ locality in classical LOCAL.

We present the first local problem that shows a super-constant separation between the classical randomized LOCAL model of distributed computing and its quantum counterpart. By prior work, such a separation was known only for an artificial graph problem with an inherently global definition [Le Gall et al. 2019]. We present a problem that we call iterated GHZ, which is defined using only local constraints. Formally, it is a family of locally checkable labeling problems [Naor and Stockmeyer 1995]; in particular, solutions can be verified with a constant-round distributed algorithm. We show that in graphs of maximum degree $Δ$, any classical (deterministic or randomized) LOCAL model algorithm will require $Ω(Δ)$ rounds to solve the iterated GHZ problem, while the problem can be solved in $1$ round in quantum-LOCAL. We use the round elimination technique to prove that the iterated GHZ problem requires $Ω(Δ)$ rounds for classical algorithms. This is the first work that shows that round elimination is indeed able to separate the two models, and this also demonstrates that round elimination cannot be used to prove lower bounds for quantum-LOCAL. To apply round elimination, we introduce a new technique that allows us to discover appropriate problem relaxations in a mechanical way; it turns out that this new technique extends beyond the scope of the iterated GHZ problem and can be used to e.g. reproduce prior results on maximal matchings [FOCS 2019, PODC 2020] in a systematic manner.

We give an almost complete characterization of the hardness of c-coloring χ-chromatic graphs with distributed algorithms, for a wide range of models of distributed computing. In particular, we show that these problems do not admit any distributed quantum advantage. To do that:

Nonlocality and its connections to entanglement are fundamental features of quantum mechanics that have found numerous applications in quantum information science. A set of correlations is said to be nonlocal if it cannot be reproduced by spacelike-separated parties sharing randomness and performing local operations. An important practical consideration is that the runtime of the parties has to be shorter than the time it takes light to travel between them. One way to model this restriction is to assume that the parties are computationally bounded. We therefore initiate the study of nonlocality under computational assumptions and derive the following results:

Parties connected to independent sources through a network can generate correlations among themselves. Notably, the space of feasible correlations for a given network, depends on the physical nature of the sources and the measurements performed by the parties. In particular, quantum sources give access to nonlocal correlations that cannot be generated classically. In this paper, we derive a universal limit on correlations in networks in terms of their covariance matrix. We show that in a network satisfying a certain condition, the covariance matrix of any feasible correlation can be decomposed as a summation of positive semidefinite matrices each of whose terms corresponds to a source in the network. Our result is universal in the sense that it holds inanyphysical theory of correlation in networks, including the classical, quantum and all generalized probabilistic theories.