Abstract. We give a complete description of the equivariant quantum cohomology ring of any smooth hypertoric variety, and find a mirror formula for the quantum differential equation.

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For any conical symplectic resolution, we give a conjecture relating the intersection cohomology of the singular cone to the quantum cohomology of its resolution. We prove this conjecture for hypertoric varieties, recovering the ring structure on hypertoric intersection cohomology that was originally constructed by Braden and the second author.

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The Hikita conjecture relates the coordinate ring of a conical symplectic singularity to the cohomology ring of a symplectic resolution of the dual conical symplectic singularity. We formulate a quantum version of this conjecture, which relates the quantized coordinate ring of the first variety to the quantum cohomology of a symplectic resolution of the dual variety. We prove this conjecture for hypertoric varieties and for the Springer resolution.moduli spaces of quasimaps from a rational curve into X! . These quasimaps, in turn, are closely related to the moduli spaces used to define X in [BFN]. We plan to explore this perspective in a future work.Remark 1.4. A different possible line of investigation is to replace the equivariant cohomology of X! by its equivariant K-theory. Then the specialized quantum D-module must be replaced by a module over difference operators, which has in many respects proved to be an even richer object [OS, AFO, Oko]. It would be interesting to see how our conjecture adapts to this setting.