Superlogarithmic Gap Result for LCLs on Trees in Quantum-LOCAL

We show that, on trees, any locally checkable labeling problem (LCL) $\mathrm{\Pi }$ that can be solved by an $n^{o(1)}$-dependent distribution can also be solved by an $O(\log n)$-round deterministic LOCAL algorithm. The result is obtained through a rake-and-compress-style decomposition of the input tree, and local simulations of the bounded dependent distribution on the components of the decomposition. As a corollary to our result, any LCL problem on trees can either be solved by an $O(\log n)$ deterministic LOCAL algorithm, or requires $n^{\mathrm{\Omega }(1)}$ rounds to solve by a quantum-LOCAL algorithm.