  We investigate the non-uniqueness of weak solutions of the
Quantum-Hydrodynamic system. This form of ill-posedness is related to the
change of the number of connected components of the support of the position
density (called nodal domains) of the weak solution throughout its time
evolution. We start by considering a scenario consisting of initial and final
time, showing that if there is a decrease in the number of connected
components, then we have non-uniqueness. This result relies on the Brouwer
invariance of domain theorem. Then we consider the case in which the results
involve a time interval and a full trajectory (position-current densities). We
introduce the concept of trajectory-uniqueness and its characterization.
