  We prove novel (local) square function/Carleson measure estimates for
non-negative solutions to the evolutionary $p$-Laplace equation in the
complement of parabolic Ahlfors-David regular sets. In the case of the heat
equation, the Laplace equation as well as the $p$-Laplace equation, the
corresponding square function estimates have proven fundamental in symmetry and
inverse/free boundary type problems, and in particular in the study of
(parabolic) uniform rectifiability. Though the implications of the square
function estimates are less clear for the evolutionary $p$-Laplace equation,
mainly due its lack of homogeneity, we give some initial applications to
parabolic uniform rectifiability, boundary behaviour and Fatou type theorems
for $\nabla_Xu$.
