  This work is concerned with the proof of \emph{a posteriori} error estimates
for fully-discrete Galerkin approximations of the Allen-Cahn equation in two
and three spatial dimensions. The numerical method comprises of the backward
Euler method combined with conforming finite elements in space. For this
method, we prove conditional type \emph{a posteriori} error estimates in the
$L^{}_4(0,T;L^{}_4(\Omega))$-norm that depend polynomially upon the inverse of
the interface length $\epsilon$. The derivation relies crucially on the
availability of a spectral estimate for the linearized Allen-Cahn operator
about the approximating solution in conjunction with a continuation argument
and a variant of the elliptic reconstruction. The new analysis also appears to
improve variants of known \emph{a posteriori} error bounds in $L_2(H^1)$,
$L_\infty^{}(L_2^{})$-norms in certain regimes.
