Matching Dependencies (MDs) are a recent proposal for declarative entity resolution. They are rules that specify, on the basis of similarities satisfied by values in a database, what values should be considered duplicates, and have to be matched. On the basis of a chase-like procedure for MD enforcement, we can obtain clean (duplicate-free), and possibly several, resolved instances. The resolved answers to a query are invariant under the class of resolved instances. Previous work identified classes of queries and sets of MDs for which resolved query answering is tractable, with special emphasis on cyclic sets of MDs. In this work we further investigate the complexity of this problem, identifying intractable cases, and exploring the frontier between tractability and intractability. We concentrate mostly on acyclic sets of MDs. For a special case we obtain a dichotomy result relative to NP-hardness.