  We prove quantum dilogarithm identities for $n$-cycle quivers. By the
combinatorial approach of Keller, each side of our identity determines a
maximal green sequence of quiver mutations. Thus we interpret our identities as
factorizations of the refined Donaldson--Thomas invariant for the quiver with
potential. Finally, we conjecture an upper bound on the possible lengths of
maximal green sequences.
