  We study the relationship between the enumerative geometry of rational curves
in local geometries and various versions of maximal contact logarithmic curve
counts. Our approach is via quasimap theory, and we show versions of the
arXiv:1712.05210 local/logarithmic correspondence for quasimaps, and in
particular for normal crossings settings, where the Gromov-Witten theoretic
formulation of the correspondence fails. The results suggest a link between
different formulations of relative Gromov-Witten theory for simple normal
crossings divisors via the mirror map. The main results follow from a rank
reduction strategy, together with a new degeneration formula for quasimaps.
