  The cluster algebra associated with an acyclic quiver has a special mutation
loop $\tau$, called the cluster Donaldson--Thomas (DT) transformation, related
to the Auslander--Reiten translation. In this paper, we characterize the
finite-tame-wild trichotomy for acyclic quivers by the sign stability of $\tau$
introduced in [IK21] and its cluster stretch factor. As an application, we
compute several kinds of entropies of $\tau$ and other mutation loops. In
particular, we show that the algebraic and categorical entropies of $\tau$ are
commonly given by the logarithm of the spectral radius of the Coxeter matrix
associated with the quiver, and that any mutation loop of finite or tame
acyclic quivers have zero algebraic entropy.
