There are several examples known of two dimensional spacetimes which are linearly stable when perturbed by test scalar classical fields, but which are unstable when perturbed by test scalar quantum fields. We elucidate the mechanism behind such instabilities by considering minimally coupled, massless, scalar, test quantum fields on general two dimensional spacetimes with Cauchy horizons which are classically stable. We identify a geometric feature of such spacetimes which is a necessary condition for obtaining a quantum mechanical divergence of the renormalized expected stress tensor on the Cauchy horizon for regular initial states. This feature is the divergence of the affine parameter length of a one parameter family of null geodesics which lie parallel to the Cauchy horizon, where the affine parameter normalization is determined by parallel transport along a fixed, transverse null geodesic which intersects the Cauchy horizon. (By contrast, the geometric feature of such spacetimes which underlies classical blueshift instabilities is the divergence of a holonomy operator). We show that the instability can be understood as a ``delayed blueshift'' instability, which arises from the infinite blueshifting of an energy flux which is created locally and quantum mechanically. The instability mechanism applies both to chronology horizons in spacetimes with closed timelike curves, and to the inner horizon in black hole spacetimes like two dimensional Reissner-Nordstrom-de Sitter.