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.

翻译：本文连接了近期探索经典分布式计算LOCAL模型扩展的三个独立研究方向：A. 分布式量子计算与非信号分布[如STOC 2024]，B. 有限相关过程[如Forum Math. Pi 2016]，以及C. 在线图算法与动态图算法中的局部性[如ICALP 2023]。针对局部可检查标记问题（LCLs），我们证明了这些计算模型在能力与局限性方面的新结果。研究表明，所有这些设定均可嵌入经典LOCAL模型与我们提出的随机化在线LOCAL模型之间。我们的工作揭示了分布式场景中量子优势的局限性，并展示了证明更紧界的新障碍。主要技术成果如下：1. 所有在经典确定性LOCAL模型中具有$O(\log^\star n)$局部性可解的LCL问题，都存在具有$O(1)$局部性的有限相关分布。这解答了Holroyd[2024]提出的开放问题，同时为基于因果性论证证明分布式量子优势界限提供了新障碍。2. 在有根树中，若能在随机化在线LOCAL模型（或任何更弱模型，如量子LOCAL）中以$o(\log \log \log n)$局部性求解LCL问题，则可在经典确定性LOCAL模型中以$O(\log^\star n)$局部性求解该问题。其重要推论之一是：在有根树中，量子LOCAL模型的$O(\log^\star n)$局部性并不强于经典LOCAL模型的$O(\log^\star n)$局部性。