  How many states of a configuration space contribute to a wave-function?
Attempts to answer this ubiquitous question have a long history in physics and
chemistry, and are keys to understand e.g. localization phenomena. Quantifying
this aspect has often been overlooked for interacting many-body quantum
systems, mainly due to the exponential growth of the configuration (Hilbert)
space. Here, we introduce two Monte Carlo schemes to calculate Shannon-Renyi
entropies for ground-states of large quantum many-body systems that are out of
reach for any other exact method. Our simulations reveal that the very nature
of quantum phases of matter and associated transitions is captured by universal
subleading terms in these entropies, on top of a generic dominant multifractal
behavior.
