  Deep connections between invariant theory and entanglement have been known
for some time and been the object of intense study. This includes the study of
local unitary equivalence of density operators as well as entanglement that can
be observed in stochastic local operations assisted by classical communication
(SLOCC). An important aspect of both of these areas is the computation of
complete sets of invariants polynomials. For local unitary equivalence as well
as $n$-qubit SLOCC invariants, complete descriptions of these invariants exist.
However, these descriptions give infinite sets; of great interest is finding
generating sets of invariants. In this regard, degree bounds are highly sought
after to limit the possible sizes of such generating sets. In this paper we
give new upper bounds on the degrees of the invariants, both for a certain
complete set of local unitary invariants as well as the $n$-qubit SLOCC
invariants. We show that there exists a complete set of local unitary
invariants of density operators in a Hilbert space $\mathcal{H}$, of dimension
$d$, which are generated by invariants of degree at most $d^4$. This in turn
allows us to show that the $n$-qubit SLOCC invariants are generated by
invariants of degree at most $2^{4n}$.
