  Current understanding of correlations and quantum phase transitions in
many-body systems has significantly improved thanks to the recent intensive
studies of their entanglement properties. In contrast, much less is known about
the role of quantum non-locality in these systems. On the one hand, standard,
"theorist- and experimentalist-friendly" many-body observables involve
correlations among only few (one, two, rarely three...) particles. On the other
hand, most of the available multipartite Bell inequalities involve correlations
among many particles. Such correlations are notoriously hard to access
theoretically, and even harder experimentally. Typically, there is no Bell
inequality for many-body systems built only from low-order correlation
functions. Recently, however, it has been shown in [J. Tura et al., Science
344, 1256 (2014)] that multipartite Bell inequalities constructed only from
two-body correlation functions are strong enough to reveal non-locality in some
many-body states, in particular those relevant for nuclear and atomic physics.
The purpose of this lecture is to provide an overview of the problem of quantum
correlations in many-body systems - from entanglement to nonlocality - and the
methods for their characterization.
