  In this paper, we prove the blowup alternative for Gross-Pitaevskii
hierarchies on $\mathbb{R}^n$ and give the associated lower bounds on the
blowup rate. In particular, we prove that any solution of density operators to
the focusing Gross-Pitaevskii hierarchy blow up in finite time for $n \ge 3$ if
the energy per some $k$ particles in the initial condition is negative. All of
these results hold without the assumption of factorized conditions for initial
values as well as the admissible ones. Our analysis is based on use of a
quasi-Banach space of sequences of marginal density matrices.
