Answer Set Programming has become a host for expressing knowledge representation problems, which reinforces the interest in efficient methods for computing answer sets of a logic program. While for propositional programs, the complexity of this task has been amply studied and is well-understood, less attention has been paid to the case of non-ground programs, which is much more important from a KR language perspective. Existing Answer Set Programming systems employ different representations of models, but the consequences of these representations for answer set computation and reasoning tasks have not been analyzed in detail. In this paper, we present novel complexity results on answer set checking for non-ground programs under two methods for representing answer sets and a variety of syntactic restrictions. In particular, we consider set-based and bitmap-based representations, which are popular in implementations of Answer Set Programming systems. Based on these results, we also derive new complexity results for the canonical reasoning tasks over answer sets, under the assumption that predicate arities are bounded by some constant. Our results imply that in such a setting - which appears to be a reasonable assumption in practice - more efficient implementations than those currently available may be feasible.