We study quantifiers and interpolation properties in emph{orthologic}, a non-distributive weakening of classical logic that is sound for formula validity with respect to classical logic and has a quadratic-time decision procedure. We present a sequent-based proof system for quantified orthologic, which we prove sound and complete for the class of all complete ortholattices. We show that orthologic does in general not admit quantifier elimination. Despite that, we show that interpolants always exists in orthologic. We give an algorithm to compute interpolants efficiently. We expect our result to be useful to quickly establish unreachability as a component of verification algorithms.