  In this paper we prove local well-posedness of a space-time fractional
generalization of the nonlinear Schr\"odinger equation with a power-type
nonlinearity. The linear equation coincides with a model proposed by Naber, and
displays a nonlocal behavior both in space and time which accounts for
long-range interactions and a so-called memory effect. Because of a loss of
derivatives produced by the latter and the lack of semigroup structure of the
solution operator, we employ a strategy of proof based on exploiting some
smoothing effect similar to that used by Kenig, Ponce and Vega for the KdV
equation. Finally, we prove analytic ill-posedness of the data-to-solution map
in the supercritical case.
