Some historical developments
A notes which explain several approaches via MSP theory and the motivation of the NMSP theory.
The key property of the NMSP theory
It behaves like the Gromov-Witten theory of a Fano variety: the generating function is a polynomial in the Novikov variable. (Theorem 1.2, [NMSP1])
An important sub-theory of the NMSP theory: the [0,1]-theory
The generating function of the [0,1]-theory has the same polynomiality as the total NMSP theory (Theorem 4, [NMSP2]). Further, it can be realized as an R-matrix action on the CohFT of the Gromov-Witten theory of the disjoint union of quintic and N points (Theorem 3, [NMSP2]).
From polynomiality of the [0,1]-theory to the BCOV's conjecture
The [0,1]-theory has many common properties with the BCOV's Feynman graph conjecture: both theory are defined via certain graph sums; the vertices in the graph both represent the Gromov-Witten theory of quintics; and the generating series for both theories are polynomials (of degree g-1 and 3g-3). Indeed, the [0,1]-theory consists of all the BCOV Feynman graphs plus some "extra" graphs. The key is that those extra graphs will contribute a polynomial of degree no more than 3g-3. ([NMSP3])