I will talk about a series of joint papers with Alexander Ritter, where we examine a large class of non-compact symplectic manifolds, including semiprojective toric varieties, conical symplectic resolutions, Higgs moduli spaces, etc.These manifolds admit a Hamiltonian circle action which is part of a pseudo-holomorphic C*-action. The symplectic form on these spaces is typically non-exact at infinity, yet we can make sense of Hamiltonian Floer cohomology for functions of the moment map of the circle action. We showed that these Floer cohomologies induce filtration by ideals on quantum cohomology, computable using certain Morse-Bott-Floer spectral sequences. I will explain recent progress on equivariant Floer cohomology for these spaces, in which case we obtain a filtration on equivariant quantum cohomology.