  We study the driven dynamics of quantum coarsening. We analyze models of
M-component rotors coupled to two electronic reservoirs at different chemical
potentials that generate a current threading through the system. In the large M
limit we derive the dynamical phase diagram as a function of temperature,
strength of quantum fluctuations, voltage and coupling to the leads. We show
that the slow relaxation in the ordering phase is universal. On large time and
length scales the dynamics are analogous to stochastic classical ones, even for
the quantum system driven out of equilibrium at zero temperature. We argue that
our results apply to generic driven quantum coarsening.
