  Nishimori's gauge theory is extended to the quantum XYZ $p$-spin glass model
in finite dimensions. This enables us to obtain useful correlation equalities,
which show also that Duhamel correlation functions at an arbitrary temperature
are bounded by those in the corresponding classical model on the Nishimori
line. These bounds give that the spontaneous magnetization vanishes in any low
temperature even if the model enters the $\mathbb Z_2$-symmetry broken spin
glass phase. This theory explains well-known fact from experiments and
numerical calculations that the magnetic susceptibility does not diverge in the
spin glass transition. The new gauge theory together with the known phase
diagram of the Edwards-Anderson model can specify the spin glass region in the
coupling constant space of the quantum Heisenberg XYZ spin glass model.
