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: <a href="http://www.ams.org/journals/proc/2004-132-10/S0002-9939-04-07274-0/S0002-9939-04-07274-0.pdf">this</a> 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.