  We develop a family of deformations of the differential and of the
pair-of-pants product on the Hamiltonian Floer complex of a symplectic manifold
(M,\omega) which upon passing to homology yields ring isomorphisms with the big
quantum homology of M. By studying the properties of the resulting deformed
version of the Oh-Schwarz spectral invariants, we obtain a Floer-theoretic
interpretation of a result of Lu which bounds the Hofer-Zehnder capacity of M
when M has a nonzero Gromov-Witten invariant with two point constraints, and we
produce a new algebraic criterion for (M,\omega) to admit a Calabi
quasimorphism and a symplectic quasi-state. This latter criterion is found to
hold whenever M has generically semisimple quantum homology in the sense
considered by Dubrovin and Manin (this includes all compact toric M), and also
whenever M is a point blowup of an arbitrary closed symplectic manifold.
