  We study the spectral statistics of quantum systems with finite Hilbert
spaces. We derive a theorem showing that eigenlevels in such systems cannot be
globally uncorrelated, even in the case of fully integrable dynamics, as a
consequence of the unfolding procedure. We provide an analytic expression for
the power spectrum of the $\delta_n$ statistic for a model of intermediate
statistics with level repulsion but independent spacings, and we show both
numerically and analytically that the result is spoiled by the unfolding
procedure. Then, we provide a simple model to account for this phenomenon, and
test it by means of numerics on the disordered XXZ chain, the paradigmatic
model of many-body localization, and the rational Gaudin-Richardson model, a
prototypical model for quantum integrability.
