Answer Set Programming with Quantifiers ASP(Q) extends Answer Set Programming (ASP) to allow for declarative and modular modeling of problems from the entire polynomial hierarchy. The first implementation of ASP(Q), called qasp, was based on a translation to Quantified Boolean Formulae (QBF) with the aim of exploiting the well-developed and mature QBF-solving technology. However, the implementation of the QBF encoding employed in qasp is very general and might produce formulas that are hard to evaluate for existing QBF solvers because of the large number of symbols and sub-clauses. In this paper, we present a new implementation that builds on the ideas of qasp and features both a more efficient encoding procedure and new optimized encodings of ASP(Q) programs in QBF. The new encodings produce smaller formulas (in terms of the number of quantifiers, variables, and clauses) and result in a more efficient evaluation process. An algorithm selection strategy automatically combines several QBF-solving back-ends to further increase performance. An experimental analysis, conducted on known benchmarks, shows that the new system outperforms qasp.