  We investigate analytically the low temperature behavior of the specific heat
$C_v(T)$ for a large class of quantum disordered models within Mean Field
approximation. This includes the vibrational modes of a lattice pinned by
impurity disorder in the quantum regime, the quantum spherical $p$-spin-glass
and a quantum Heisenberg spin glass. We exhibit a general mechanism, common to
all these models, arising from the so-called marginality condition, responsible
for the cancellation of the linear and quadratic contributions in $T$ in the
specific heat. We thus find for all these models the Mean Field result $C_v(T)
\propto T^3$.

Disordered Systems and Neural Networks