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We consider the 3D quantum BBGKY hierarchy which corresponds to the$N$-particle Schr\"{o}dinger equation. We assume the pair interaction is$N^{3\beta -1}V(N^{\beta}\bullet).$ For interaction parameter $\beta\in(0,\frac23)$, we prove that, as $N\rightarrow \infty ,$ the limit points ofthe solutions to the BBGKY hierarchy satisfy the space-time bound conjecturedby Klainerman-Machedon in 2008. This allows for the application of theKlainerman-Machedon uniqueness theorem, and hence implies that the limit isuniquely determined as a tensor product of solutions to the Gross-Pitaevskiequation when the $N$-body initial data is factorized. The first result in thisdirection in 3D was obtained by T. Chen and N. Pavlovi\'{c} (2011) for $\beta\in (0,\frac14)$ and subsequently by X. Chen (2012) for $\beta\in (0,\frac27]$.We build upon the approach of X. Chen but apply frequency localizedKlainerman-Machedon collapsing estimates and the endpoint Strichartz estimatein the estimate of the potential part to extend the range to $\beta\in(0,\frac23)$. Overall, this provides an alternative approach to the mean-fieldprogram by Erd\"os-Schlein-Yau (2007), whose uniqueness proof is based uponFeynman diagram combinatorics.

On the Klainerman-Machedon Conjecture of the Quantum BBGKY Hierarchy with Self-interaction