  We establish virial and localized virial identities for solutions to the
Hartree hierarchy, an infinite system of partial differential equations which
arises in mathematical modeling of many body quantum systems. As an
application, we use arguments originally developed in the study of the
nonlinear Schr\"odinger equation (see work of Zakharov, Glassey, and
Ogawa--Tsutsumi) to show that certain classes of negative energy solutions must
blow up in finite time.
  The most delicate case of this analysis is the proof of negative energy
blowup without the assumption of finite variance; in this case, we make use of
the localized virial estimates, combined with the quantum de Finetti theorem of
Hudson and Moody and several algebraic identities adapted to our particular
setting. Application of a carefully chosen truncation lemma then allows for the
additional terms produced in the localization argument to be controlled.
