Superpolynomial Simulator
Well, your edited version of the question revealed a reference to the following paper:

Rafael Pass. 2003. Simulation in quasi-polynomial time, and its application to protocol composition. In Proceedings of the 22nd international conference on Theory and applications of cryptographic techniques (EUROCRYPT'03), Eli Biham (Ed.). Springer-Verlag, Berlin, Heidelberg, 160-176. (See also this version.)

The paper discusses and justifies the need for such simulators. 
The abstract reads:


  We propose a relaxation of zero-knowledge, by allowing the
  simulator to run in quasi-polynomial time. We show that protocols satisfying this notion can be constructed in settings where the standard definition is too restrictive. Specifically, we construct constant-round straight-line concurrent quasi-polynomial time simulatable arguments and show
  that such arguments can be used in advanced composition operations without any set-up assumptions. Our protocols rely on slightly strong,
  but standard type assumptions (namely the existence of one-to-one one-way functions secure against subexponential circuits).


For a more thorough discussion, I suggest taking a look at Rafael Pass's M.Sc. thesis.