  This article consists of two parts. In the first part, we review the most
recent proofs establishing quadratic Morawetz inequalities for the nonlinear
Schr\"odinger equation (NLS). We also describe the applications of these
estimates to the problem of quantum scattering. In the second part, we
generalize some of the methods developed for the NLS by many authors to the
case of Gross-Pitaevskii (GP) hierarchies. In particular, we prove both regular
and interaction Morawetz identities for the GP hierarchy, which appear here for
the first time in the literature.
