We extend the theory of locally checkable labeling problems (LCLs) from the classical LOCAL model to a number of other models that have been studied recently, including the quantum-LOCAL model, finitely-dependent processes, non-signaling model, dynamic-LOCAL model, and online-LOCAL model [e.g. STOC 2024, ICALP 2023]. First, we demonstrate the advantage that finitely-dependent processes have over the classical LOCAL model. We show that all LCL problems solvable with locality $O(\log^* n)$ in the LOCAL model admit a finitely-dependent distribution (with constant locality). In particular, this gives a finitely-dependent coloring for regular trees, answering an open question by Holroyd [2023]. This also introduces a new formal barrier for understanding the distributed quantum advantage: it is not possible to exclude quantum advantage for any LCL in the $\Theta(\log^* n)$ complexity class by using non-signaling arguments. Second, we put limits on the capabilities of all of these models. To this end, we introduce a model called randomized online-LOCAL, which is strong enough to simulate e.g. SLOCAL and dynamic-LOCAL, and we show that it is also strong enough to simulate any non-signaling distribution and hence any quantum-LOCAL algorithm. We prove the following result for trees: if we can solve an LCL problem with locality $o(\log^{(5)} n)$ in the randomized online-LOCAL model, we can solve it with locality $O(\log^* n)$ in the classical deterministic LOCAL model. Put together, these results show that in trees the set of LCLs that can be solved with locality $O(\log^* n)$ is the same across all these models: locality $O(\log^* n)$ in quantum-LOCAL, non-signaling model, dynamic-LOCAL, or online-LOCAL is not stronger than locality $O(\log^* n)$ in the classical deterministic LOCAL model.

翻译：我们将局部可检查标记问题（LCLs）的理论从经典LOCAL模型扩展至近期研究的多种其他模型，包括量子-LOCAL模型、有限依赖过程、非信号模型、动态-LOCAL模型和在线-LOCAL模型[例如STOC 2024, ICALP 2023]。首先，我们展示了有限依赖过程相对于经典LOCAL模型的优势。我们证明，所有在LOCAL模型中具有局部性$O(\log^* n)$的可解LCL问题，都允许一个有限依赖分布（具有恒定局部性）。特别地，这为正则树提供了有限依赖着色，回答了Holroyd [2023]的一个开放问题。这也为理解分布式量子优势引入了一个新的形式化障碍：无法通过非信号论证排除$\Theta(\log^* n)$复杂度类中任何LCL的量子优势。其次，我们对所有这些模型的能力施加了限制。为此，我们引入了一个称为随机化在线-LOCAL的模型，它足够强大以模拟如SLOCAL和动态-LOCAL等模型，并且我们证明它也足够强大以模拟任何非信号分布，从而模拟任何量子-LOCAL算法。我们针对树证明了如下结果：如果我们在随机化在线-LOCAL模型中能够解决局部性为$o(\log^{(5)} n)$的LCL问题，那么我们在经典确定性LOCAL模型中就能解决局部性为$O(\log^* n)$的相同问题。综合这些结果表明，在树中，所有模型下可解且局部性为$O(\log^* n)$的LCL集合是相同的：量子-LOCAL、非信号模型、动态-LOCAL或在线-LOCAL中的局部性$O(\log^* n)$并不强于经典确定性LOCAL模型中的局部性$O(\log^* n)$。