  We study essentially bounded quantum random variables and show that the
Gelfand spectrum of such a quantum random variable coincides with the
hypoconvex hull of its essential range. Moreover, a notion of operator-valued
variance is introduced, leading to a formulation of the moment problem in the
context of quantum probability spaces in terms of operator-theoretic properties
involving semi-invariant subspaces and spectral theory. As an application of
quantum variance, new measures of random and inherent quantum noise are
introduced for measurements of quantum systems, modifying some recent ideas of
Polterovich.
