A homological condition that might be useful: in the Hopf case, the Yoneda algebra $Ext_A^\bullet(k,k)$ embeds into the Hochschild cohomology $HH^\bullet(A,A)$, moreover, there is a Gerstenhaber algebra structure on the Yoneda algebra, and this embedding is an embedding of Gerstenhaber algebras.
Reference: this article of Marco Farinati and Andrea Solotar.
I have a feeling that it would give some information already for exterior algebras, though I don't have time to check it carefully now.