  We study state-sum constructions of G-equivariant spin-TQFTs and their
relationship to Matrix Product States. We show that in the Neveu-Schwarz,
Ramond, and twisted sectors, the states of the theory are generalized Matrix
Product States. We apply our results to revisit the classification of fermionic
Short-Range-Entangled phases with a unitary symmetry G and determine the group
law on the set of such phases. Interesting subtleties appear when the total
symmetry group is a nontrivial extension of G by fermion parity.
