  We prove the existence of global in time, finite energy, weak solutions to a
quantum magnetohydrodynamic system (QMHD) with large data, modeling a charged
quantum fluid interacting with a self-generated electromagnetic field. The
analysis of QMHD relies upon the use of Madelung transformations. The rigorous
derivation requires non-trivial smoothing estimates, which are obtained by
assuming slightly higher regularity for the electromagnetic potential. These
assumptions are motivated by the nonlinear dependence of the hydrodynamic
system in terms of the underlying wave function dynamics, which is
supercritical with respect to the bare energy bounds. Due to quantum effects on
the dispersive properties of QMHD, our approach requires neither smallness nor
high regularity, unlike a large amount of existing literature for
Euler-Maxwell's classical system. For quantum MHD system the irrotationality
and the presence of a highly nonlinear quantum stress tensor induce much
stronger dispersive properties, as a byproduct of a close relationship with the
classical Maxwell-Schr\"odinger system. Therefore the core argument is shifted
to the analysis of the nonlinearities related to the formulation of the
hydrodynamic variables through the Madelung transformations. The analysis
carried out in section 4 shows that it is necessary to go through non-trivial
smoothing estimates and these require us to assume regularity conditions, just
above the energy norms, for the initial data of the Maxwellian electromagnetic
potential. In the same regime of regularity, with the help of suitable local
smoothing estimates, we also prove stability of both the hydrodynamic variables
and the Lorentz force associated with the electromagnetic field. (Abbreviated
version, see full abstract in the paper).
