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：我们给出对使用分布式算法为χ-色图进行c-着色的难解性近乎完整的刻画，涵盖一系列广泛的分布式计算模型。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）我们提出一种新的分布式算法，可在$\tilde{\mathcal{O}}(n^{\frac{1}{\alpha}})$轮内为χ-色图找到c-着色，其中$\alpha = \bigl\lfloor\frac{c-1}{\chi - 1}\bigr\rfloor$。2）我们证明任何解决该问题的分布式算法都需要$\Omega(n^{\frac{1}{\alpha}})$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在非信令模型中成立。该模型由Arfaoui和Fraigniaud于2014年提出，囊括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，还涵盖量子-LOCAL（即使预先共享量子态）。我们还表明，类似论证可证明例如：对二维网格进行3-着色或对树进行c-着色，在非信令模型中仍为难题，且特别地不存在任何量子优势。我们的下界论证本质上是纯图论性质的；无需量子信息论背景即可理解证明过程。