  In this paper, we study the Hermitian hulls of (extended) generalized
Reed-Solomon (GRS and EGRS) codes over finite fields. For a given class of
(extended) GRS codes, by increasing the length, increasing the dimensions and
increasing both the length and the dimensions, we obtain three new classes of
(extended) GRS codes with Hermitian hulls of arbitrary dimensions. Furthermore,
we obtain several new classes of $q^2$-ary maximum distance separable (MDS)
codes with Hermitian hulls of arbitrary dimensions. And the dimension of these
MDS codes can be taken from $1$ to $\frac{n}{2}$. By propagation rules, the
parameters of the obtained code can be more flexible. As an application, a lot
of new (MDS) entanglement-assisted quantum error correction codes (EAQECCs) can
be constructed from previous known (extended) GRS codes. We derive three new
propagation rules on (MDS) EAQECCs constructed from (extended) GRS codes.
Finally, we present several new classes of (MDS) EAQECCs with flexible
parameters. Notably, the distance parameters of our codes can range from $2$ to
$\frac{n+2}{2}$.
