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：我们联系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[例如STOC 2024]，B. 有限相关过程[例如Forum Math. Pi 2016]，以及C. 在线图算法与动态图算法中的局部性[例如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。我们证明所有这些设定都可以被夹在经典LOCAL模型与我们称之为随机化在线-LOCAL模型之间。我们的工作暗示了分布式场景中量子优势的局限性，同时我们也展示了一个证明更紧界限的新障碍。我们的主要技术成果如下：1. 所有在经典确定性LOCAL模型中可通过$O(\log^\star n)$局部性求解的LCL问题，都存在具有$O(1)$局部性的有限相关分布。这回答了Holroyd[2024]提出的一个开放性问题，同时也为使用基于因果性的论证来证明分布式量子优势的界限提出了新障碍。2. 在有根树中，如果我们能在随机化在线-LOCAL模型（或任何更弱的模型，如量子-LOCAL）中以$o(\log \log \log n)$局部性求解一个LCL问题，那么我们就能在经典确定性LOCAL模型中以$O(\log^\star n)$局部性求解它。众多推论之一是：在有根树中，量子-LOCAL模型中的$O(\log^\star n)$局部性并不强于经典LOCAL模型中的$O(\log^\star n)$局部性。