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We introduce the notion of quantum-symmetric equivalence of two connectedgraded algebras, based on Morita--Takeuchi equivalences of their universalquantum groups, in the sense of Manin. We study homological and algebraicinvariants of quantum-symmetric equivalence classes, and in particular provethat numerical $\mathrm{Tor}$-regularity, Castelnuovo--Mumford regularity,Artin--Schelter regularity, and the Frobenius property are invariant under anyMorita--Takeuchi equivalence. We characterize 2-cocycle twists (which arise asa special case of quantum-symmetric equivalence) of Koszul duals,superpotentials and superpotential algebras, Nakayama automorphisms of twistedFrobenius algebras, and Artin--Schelter regular algebras. We also show thatfinite generation of Hochschild cohomology rings is preserved under certain2-cocycle twists.

Twisting Manin's universal quantum groups and comodule algebras