  In this paper we prove the global in time well-posedness of strong solutions
to the Quantum-Navier-Stokes equation driven by random initial data and
stochastic external force. In particular, we first give a general local
well-posedness result. Then, by means of the Bresch-Desjardins entropy, higher
order energy estimates, and a continuation argument we prove that the density
never vanishes, and thus that local strong solutions are indeed global.
