  In this paper, we study solutions to the three-dimensional quintic
Gross-Pitaevskii hierarchy. We prove unconditional uniqueness among all small
solutions in the critical space $\mathfrak{H}^1$ (which corresponds to $H^1$ on
the NLS level). With slight modifications to the proof, we also prove
unconditional uniqueness of solutions to the Hartree hierarchy without
smallness condition. Our proof uses the quantum de Finetti theorem, and is an
extension of the work by Chen-Hainzl-Pavlovi\'c-Seiringer \cite{CHPS}, and our
previous work \cite{UniqueLowReg}.
