  We study the problem of unconditional uniqueness of solutions to the cubic
nonlinear Schr\"odinger equation. We introduce a new strategy to approach this
problem on bounded domains, in particular on rectangular tori.
  It is a known fact that solutions to the cubic NLS give rise to solutions of
the Gross-Pitaevskii hierarchy, which is an infinite-dimensional system of
linear equations. By using the uniqueness analysis of the Gross-Pitaevskii
hierarchy, we obtain new unconditional uniqueness results for the cubic NLS on
rectangular tori, which cover the full scaling-subcritical regime in high
dimensions. In fact, we prove a more general result which is conditional on the
domain.
  In addition, we observe that well-posedness of the cubic NLS in
Fourier-Lebesgue spaces implies unconditional uniqueness.
