  We define two invariants for (semiprime right Goldie) algebras, one for
algebras graded by arbitrary abelian groups, which is unchanged under twists by
$2$-cocycles on the grading group, and one for $\mathbb Z$-graded or $\mathbb
Z_{\ge 0}$-filtered algebras. The first invariant distinguishes quantum
algebras which are "truly multiparameter" apart from ones that are "essentially
uniparameter", meaning cocycle twists of uniparameter algebras. We prove that
both invariants are stable under adjunction of polynomial variables. Methods
for computing these invariants for large families of algebras are given,
including quantum nilpotent algebras and algebras admitting one quantum
cluster, and applications to non-isomorphism theorems are obtained.
