  In frustrated quantum magnets the empirically found quantum-to-classical
correspondence (QCC) matches the real-space static susceptibility pattern of a
quantum spin-$1/2$ model with its classical counterpart computed at a certain
elevated temperature. This puzzling relation was observed via bold line
diagrammatic Monte Carlo simulations in dimensions two and three, where the
matching was within error bars and seemed valid down to the lowest accessible
temperatures $T$ about an order of magnitude smaller than the exchange coupling
$J$. Here we employ resummed spin diagrammatic perturbation theory to show
analytically that the QCC breaks at fourth order in $J/T$ and provide the
approximate mapping between classical and quantum temperatures. Our treatment
further reveals that QCC is an indication of the surprising accuracy with which
static correlators can be approximated by a simple renormalized mean-field
form. We illustrate this for all models discussed in the context of QCC so far,
including a recent example of the $S=1$ material
$\mathrm{K}_2\mathrm{Ni}_2(\mathrm{SO}_4)_3$. The success of the mean-field
form is traced back to partial diagrammatic cancellations.
