  We construct a Convolution Quadrature (CQ) scheme for the quasilinear
subdiffusion equation and supply it with the fast and oblivious implementation.
In particular we find a condition for the CQ to be admissible and discretize
the spatial part of the equation with the Finite Element Method. We prove the
unconditional stability and convergence of the scheme and find a bound on the
error. As a passing result, we also obtain a discrete Gronwall inequality for
the CQ, which is a crucial ingredient of our convergence proof based on the
energy method. The paper is concluded with numerical examples verifying
convergence and computation time reduction when using fast and oblivious
quadrature.
