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.

<s>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.</s> Of course, to use this observation for exterior algebras, the graded commutative product from the Gerstenhaber structure (highlighted by [mt](http://mathoverflow.net/users/6481/mt) in his answer) is enough. But I think that there are cases where the Lie bracket will help to settle the answer.