Quantum lower bounds for 3-coloring cycles in the quantum-LOCAL model

Establish nontrivial quantum lower bounds on the round complexity of 3-coloring cycles in the quantum-LOCAL model, using techniques other than round elimination, which the paper identifies as inapplicable for this purpose.

Background

The authors prove that round elimination can separate classical LOCAL from quantum-LOCAL for their iterated GHZ problem, but also show that round elimination does not apply to quantum-LOCAL generally, creating a formal barrier.

They highlight that resolving major open questions, such as proving quantum lower bounds for 3-coloring cycles, cannot proceed via round elimination, motivating the development of new techniques.

References

This also represents a new formal barrier for resolving major open questions related to distributed quantum advantage: it is not possible to prove quantum lower bounds for e.g. 3-coloring cycles or for finding sinkless orientations by constructing a round elimination sequence.

— Distributed Quantum Advantage for Local Problems  (2411.03240 - Balliu et al., 2024) in Implications and discussion, Lower-bound proof techniques paragraph

The number of colors in \cref{res:main-epsilon} is large. By we know that it has to be more than $4$, but the precise value is unknown: What is the smallest $c$ such that for any $\epsilon > 0$ we can find a $c$-coloring in directed cycles with one-way one-round quantum algorithms so that for each edge the local failure probability is at most $\epsilon$?

— Quantum Advantage for Distributed Symmetry Breaking  (2609.26788 - Flin et al., 22 Sep 2026) in Section "Open Questions", third problem statement