  We present a quantum Monte Carlo study of the "quantum glass" phase of the 2D
Bose-Hubbard model with random potentials at filling $\rho=1$. In the narrow
region between the Mott and superfluid phases the compressibility has the form
$\kappa \sim {\rm exp}(-b/T^\alpha)+c$ with $\alpha <1$ and $c$ vanishing or
very small. Thus, at $T=0$ the system is either incompressible (a Mott glass)
or nearly incompressible (a Mott-glass-like anomalous Bose glass). At stronger
disorder, where a glass reappears from the superfluid, we find a conventional
highly compressible Bose glass. On a path connecting these states, away from
the superfluid at larger Hubbard repulsion, a change of the disorder strength
by only $10\%$ changes the low-temperature compressibility by more than four
orders of magnitude, lending support to two types of glass states separated by
a phase transition or a sharp cross-over.
