We give an almost complete characterization of the hardness of $c$-coloring $\chi$-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: 1) We give a new distributed algorithm that finds a $c$-coloring in $\chi$-chromatic graphs in $\tilde{\mathcal{O}}(n^{\frac{1}{\alpha}})$ rounds, with $\alpha = \bigl\lfloor\frac{c-1}{\chi - 1}\bigr\rfloor$. 2) We prove that any distributed algorithm for this problem requires $\Omega(n^{\frac{1}{\alpha}})$ rounds. Our upper bound holds in the classical, deterministic LOCAL model, while the near-matching lower bound holds in the non-signaling model. This model, introduced by Arfaoui and Fraigniaud in 2014, captures all models of distributed graph algorithms that obey physical causality; this includes not only classical deterministic LOCAL and randomized LOCAL but also quantum-LOCAL, even with a pre-shared quantum state. We also show that similar arguments can be used to prove that, e.g., 3-coloring 2-dimensional grids or $c$-coloring trees remain hard problems even for the non-signaling model, and in particular do not admit any quantum advantage. Our lower-bound arguments are purely graph-theoretic at heart; no background on quantum information theory is needed to establish the proofs.

翻译：我们给出了对于分布式算法在广泛计算模型下对$\chi$-色图进行$c$-着色问题难度的几乎完整刻画。特别地，我们证明这些问题不存在任何分布式量子优势。为此：1) 提出一种新的分布式算法，可在$\tilde{\mathcal{O}}(n^{\frac{1}{\alpha}})$轮内为$\chi$-色图找到$c$-着色，其中$\alpha = \bigl\lfloor\frac{c-1}{\chi - 1}\bigr\rfloor$。2) 证明任何解决该问题的分布式算法至少需要$\Omega(n^{\frac{1}{\alpha}})$轮。我们的上界适用于经典确定性LOCAL模型，而近乎匹配的下界则适用于非信令模型。该模型由Arfaoui和Fraigniaud于2014年提出，囊括了所有遵循物理因果律的分布式图算法模型，不仅包括经典确定性LOCAL和随机化LOCAL，还包括量子-LOCAL（即使预共享量子态）。我们还证明类似论证可用于表明：例如二维网格的3-着色或树的$c$-着色等难题，对非信令模型同样成立，且尤其不存在量子优势。我们的下界论证本质上完全基于图论，无需量子信息论背景即可完成证明。