Marklof and Rudnick proved that for any rational polygon, a density-one subsequence of Laplace eigenfunctions equidistributes in configuration space, despite the absence of ergodicity of the full billiard flow in phase space. (If the flow were truly ergodic, equidistribution would follow from an extension by Zelditch and Zworski of the well known quantum ergodicity theorem to manifolds with boundary). I will discuss a quantitative refinement of the Marklof-Rudnick theorem and present a polynomial rate of decay for the quantum variance with respect to the frequency parameter. This leads to new results on small scale equidistribution and improved L^p estimates for Laplace eigenfunctions on rational polygons. The proofs use new estimates for the classical rate of ergodicity and some microlocal analysis to deal with conical singularities.