  We consider quantum-dynamical phenomena in the $\mathrm{SU}(2)$, $S=1/2$
infinite-range quantum Heisenberg spin glass. For a fermionic generalization of
the model we formulate generic dynamical self-consistency equations. Using the
Popov-Fedotov trick to eliminate contributions of the non-magnetic fermionic
states we study in particular the isotropic model variant on the spin space.
Two complementary approximation schemes are applied: one restricts the quantum
spin dynamics to a manageable number of Matsubara frequencies while the other
employs an expansion in terms of the dynamical local spin susceptibility. We
accurately determine the critical temperature $T_c$ of the spin glass to
paramagnet transition. We find that the dynamical correlations cause an
increase of $T_c$ by 2% compared to the result obtained in the spin-static
approximation. The specific heat $C(T)$ exhibits a pronounced cusp at $T_c$.
Contradictory to other reports we do not observe a maximum in the $C(T)$-curve
above $T_c$.
