  We study a quantum version of the three-state Potts model that includes as
special cases the effective models of bosons and fermions on the square lattice
in the Mott insulating limit. It can be viewed as a model of quantum
permutations with amplitudes J_parallel and J_perp for identical and different
colors, respectively. For J_parallel=J_perp>0, it is equivalent to the SU(3)
Heisenberg model, which describes the Mott insulating phase of 3-color
fermions, while the parameter range J_perp<min(0,-J_parallel) can be realized
in the Mott insulating phase of 3-color bosonic atoms. Using linear flavor wave
theory, infinite projected entangled-pair states (iPEPS), and continuous-time
quantum Monte-Carlo simulations, we construct the full T=0 phase diagram, and
we explore the T>0 properties for J_perp<0. For dominant antiferromagnetic
J_parallel interactions, a three-sublattice long-range ordered stripe state is
selected out of the ground state manifold of the antiferromagnetic Potts model
by quantum fluctuations. Upon increasing |J_perp|, this state is replaced by a
uniform superfluid for J_perp<0, and by an exotic three-sublattice superfluid
followed by a two-sublattice superfluid for J_perp>0. The transition out of the
uniform superfluid (that can be realized with bosons) is shown to be a peculiar
type of Kosterlitz-Thouless transition with three types of elementary vortices.
