In this paper, we provide a comprehensive understanding of the complexity of $c$-coloring $chi$-chromatic graphs using distributed algorithms in various models of distributed computing. We demonstrate that these problems do not offer any distributed quantum advantage. Firstly, we present a new distributed algorithm that can identify a $c$-coloring in $chi$-chromatic graphs in approximately $tilde{mathcal{O}}(n^{frac{1}{alpha}})$ rounds. Here, $alpha = bigllfloorfrac{c-1}{chi – 1}bigrrfloor$. Secondly, we establish that any distributed algorithm for this problem necessitates at least $Omega(n^{frac{1}{alpha}})$ rounds. Our upper bound applies to the classical, deterministic LOCAL model, while the nearly matching lower bound applies to the non-signaling model. This model, introduced by Arfaoui and Fraigniaud in 2014, encompasses all distributed graph algorithm models that abide by physical causality. It includes classical deterministic LOCAL, randomized LOCAL, quantum-LOCAL with a pre-shared quantum state. We also prove that similar arguments can demonstrate the hardness of other problems,
https://arxiv.org/abs/2307.09444