  Online contention resolution schemes (OCRSs) were proposed by Feldman,
Svensson, and Zenklusen as a generic technique to round a fractional solution
in the matroid polytope in an online fashion. It has found applications in
several stochastic combinatorial problems where there is a commitment
constraint: on seeing the value of a stochastic element, the algorithm has to
immediately and irrevocably decide whether to select it while always
maintaining an independent set in the matroid. Although OCRSs immediately lead
to prophet inequalities, these prophet inequalities are not optimal. Can we
instead use prophet inequalities to design optimal OCRSs?
  We design the first optimal $1/2$-OCRS for matroids by reducing the problem
to designing a matroid prophet inequality where we compare to the stronger
benchmark of an ex-ante relaxation. We also introduce and design optimal
$(1-1/e)$-random order CRSs for matroids, which are similar to OCRSs but the
arrival is chosen uniformly at random.
