Nonnegative weak solutions of quasi-linear degenerate parabolic equations of $p$ -Laplacian type are shown to be locally bounded below by Barenblatt-type subpotentials. As a consequence, nonnegative solutions expand their positivity set. That is, a quantitative lower bound on a ball $B_\rho$ at time $\bar{t}$ yields a quantitative lower bound on a ball $B_{2\rho}$ at some further time $t$ . These lower bounds also permit one to recast the Harnack inequality of [4] in a family of alternative, equivalent forms