By a result of Kadin, the difference hierarchy over NP does not collapse if the polynomial hierarchy does not. We extend this result to a natural refinement of the polynomial hierarchy. The refinement is obtained from levels of the polynomial hierarchy by validly applying addition modulo 2, and so here we call it the plus-hierarchy. Also, we consider two refinements of the plus-hierarchy, which may be relevant to a classification of some languages. © 1999 Kluwer Academic/Plenum Publishers.