  We introduce the notion of quantum-symmetric equivalence of two connected
graded algebras, based on Morita-Takeuchi equivalences of their universal
quantum groups, in the sense of Manin. We study homological and algebraic
invariants of quantum-symmetric equivalence classes, and prove that numerical
$\mathrm{Tor}$-regularity, Castelnuovo-Mumford regularity, Artin-Schelter
regularity, and the Frobenius property are invariant under any Morita-Takeuchi
equivalence. In particular, by combining our results with the work of
Raedschelders and Van den Bergh, we prove that Koszul Artin-Schelter regular
algebras of a fixed global dimension form a single quantum-symmetric
equivalence class. Moreover, we characterize 2-cocycle twists (which arise as a
special case of quantum-symmetric equivalence) of Koszul duals, of
superpotentials, of superpotential algebras, of Nakayama automorphisms of
twisted Frobenius algebras, and of Artin-Schelter regular algebras. We also
show that finite generation of Hochschild cohomology rings is preserved under
certain 2-cocycle twists.
