  In this review, we analyse different aspects concerning the possibility to
separate a gravity-matter system into a part which lives close to a
quasi-classical state and a "small" quantum subset. The considered approaches
are all relying on a WKB expansion of the dynamics by an order parameter and
the natural arena consists of the Bianchi universe minisuperspace. We first
discuss how, limiting the WKB expansion to the first order of approximation, it
is possible to recover for the quantum subsystem a Schr\"odinger equation as
written on the classical gravitational background. We give some applications
for the quantum subsystem in the "corner" configuration of the Bianchi IX
model, identifying the quantum variable in the small one of the two anisotropy
degrees of freedom. The most surprising result is the possibility to get a
non-singular Bianchi IX cosmology when the scenario is extrapolated backwards
in time. We also provide some basic hints on the extension of this result to
the generic cosmological solution. In the last part of the review, we consider
the same scheme to the next order of approximation identifying the quantum
subset as made of matter variables only. We deal with the very fundamental
problem of non-unitary morphology of the quantum gravity corrections to quantum
field theory discussing some proposed reformulations. Instead of constructing
the time dependence via that one of the classical gravitational variables on
the label time, we analyse a recent proposal to construct time by fixing a
reference frame. This scheme can be reached both introducing the so-called
"kinematical action", as well as by the well-known Kuchar-Torre formulation. In
both cases, the Schr\"odinger equation, amended for quantum gravity
corrections, has the same morphology and we provide a cosmological
implementation of the model, to elucidate its possible predictions.
