Unconditional Uniqueness for the Energy-critical and Energy-supercritical Quadratic NLS

We study the quadratic nonlinear Schr\"{o}dinger equation in energy-critical and energy-supercritical regimes and establish unconditional uniqueness at critical regularity on both $\mathbb{T}^{d}$ and $\mathbb{R}^{d}$. We introduce a new infinite quadratic hierarchy that featuring a linear structure for tensor product forms, instead of marginal densities. Consequently, this newly constructed hierarchy differs from the quantum Gross-Pitaevskii hierarchy, and its structure is in fact closer to that of the classical Boltzmann hierarchy. We prove this quadratic hierarchy admits combinatorial structures that are compatible with the bilinear $U$-$V$ estimates we prove for the quadratic nonlinearity. These tools enable us to establish unconditional uniqueness at the critical regularity via a quadratic hierarchy approach, and thus to provide an affirmative answer to Bourgain's uniqueness concern [3,p.152] for the quadratic nonlinearity in the weak-form bilinear case.