Abstract: The geometry of toric Fano varieties has been studied through the combinatorics of their fans and polytopes. Introduced by Batyrev, reflexive polytopes are lattice polytopes whose dual polytope is also a lattice polytope. There are finitely many reflexive polytopes in each dimension up to isomorphism, and they correspond to the toric Fano varieties with Gorenstein singularities. There have been various approaches to estimate the Picard ranks of Gorenstein toric Fano varieties. In the Q-factorial case, the Picard ranks are at most twice the dimension, while in general an upper bound remains unknown. I will report a work in progress, where we introduce new methods estimating the Picard ranks, with applications in the case of terminal singularities in dimension 3 and 4. This generalizes previous works from Casagrande, Eikelberg, Fujita and Nill.