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 $\alpha = \bigl\lceil\frac{c-1}{\chi - 1}\bigr\rceil$. 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 \emph{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：我们几乎完整刻画了在分布式计算的各种模型下，用分布式算法对χ-色图进行c-着色的难度。特别地，我们证明这些问题不具有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，能在χ-色图中以$\tilde{\mathcal{O}}(n^{\frac{1}{\alpha}})$轮次找到c-着色，其中$\alpha = \bigl\lceil\frac{c-1}{\chi - 1}\bigr\rceil$。2. 我们证明任何解决此问题的分布式算法都需要$\Omega(n^{\frac{1}{\alpha}})$轮次。我们的上界适用于经典确定性LOCAL模型，而近乎匹配的下界适用于*非信令*模型。该模型由Arfaoui和Fraigniaud于2014年提出，涵盖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，类似的论证可用于证明，例如，2维网格的3-着色或树的c-着色即使在非信令模型中也难以解决，因此不具有任何量子优势。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建立证明。