The Topological String/Spectral Theory (TS/ST) correspondence was introduced as a sharp, non-perturbative relationship between topological strings on toric Calabi-Yau threefolds on one side, and the spectral theory of operators given by the quantized mirror curve on the other side. It predicts a non-trivial relationship between enumerative invariants of the toric Calabi-Yau threefold and spectral quantities of the corresponding operator. This work explores three consequences and extensions of the TS/ST correspondence. First, we show that the TS/ST correspondence implies a new realization of the topological string as convergent matrix models. Second, we propose an extension of the TS/ST correspondence involving the open sector of topological strings on the TS side and the eigenfunctions of the operator on the ST side. Third, we investigate how the TS/ST correspondence and its interpretation in terms of a non-interacting Fermi gas can help us defining a sensible notion of "quantum curve".