Temporal Equilibrium Logic (TEL) is a promising framework that extends the knowledge representation and reasoning capabilities of Answer Set Programming with temporal operators in the style of LTL. To our knowledge it is the first nonmonotonic logic that accommodates fully the syntax of a standard temporal logic (specifically LTL) without requiring further constructions. This paper provides a systematic complexity analysis for the (consistency) problem of checking the existence of a temporal equilibrium model of a TEL formula. It was previously shown that this problem in the general case lies somewhere between PSPACE and EXPSPACE. Here we establish a lower bound matching the known EXPSPACE upper bound. Additionally we analyse the complexity for various natural subclasses of TEL formulas, identifying both tractable and intractable fragments. Finally the paper offers some new insights on the logic LTL by addressing satisfiability for minimal LTL models. The complexity results obtained highlight a substantial difference between interpreting LTL over finite or infinite words.