We present a distributed quantum algorithm that $3$-colors cycles in $O(1)$ rounds, with high probability. It follows that all locally checkable labeling problems (LCLs) that have round complexity $O(\log^* n)$ in the classical LOCAL model can be solved in $O(1)$ rounds in the quantum-LOCAL model, with high probability; this includes problems such as maximal independent set and maximal matching in bounded-degree graphs. This presents the first natural examples of graph problems with an asymptotic distributed quantum advantage for the LOCAL model; all prior examples that separate LOCAL and quantum-LOCAL are artificial problems constructed merely for the sake of demonstrating quantum advantage.
We show that one-way one-round quantum-LOCAL algorithms cannot $4$-color directed cycles with high probability. This is the first lower bound in the high-probability quantum LOCAL setting that goes beyond the non-signaling and bounded-dependence models, exploiting the structure of distributed quantum algorithms. Our proof establishes a bidirectional connection between distributed quantum computing and extremal combinatorics. We obtain our lower bound by proving a Mantel-type stability theorem for weighted matrix spaces.