Local regularity for quasi-linear parabolic equations in non-divergence form

We consider viscosity solutions to non-homogeneous degenerate and singular parabolic equations of the $p$-Laplacian type and in non-divergence form. We provide local Hölder and Lipschitz estimates for the solutions. In the degenerate case, we prove the Hölder regularity of the gradient. Our study is based on a combination of the method of alternatives and the improvement of flatness estimates.