Uniqueness of solutions to the 3D quintic Gross-Pitaevskii Hierarchy

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}.