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mCP-nets are an expressive and intuitive formalism based on CP-nets to reason about preferences of groups of agents. The dominance semantics of mCP-nets is based on the concept of voting, and different voting schemes give rise to different dominance semantics for the group. Unlike CP-nets, which received an extensive complexity analysis, mCP-nets, as reported multiple times in the literature, lack a precise study of the voting tasks' complexity. Prior to this work, only a complexity analysis of brute-force algorithms for these tasks was available, and this analysis only gave EXPTIME upper bounds for most of those problems. In this paper, we start to fill this gap by carrying out a precise computational complexity analysis of voting tasks on acyclic binary polynomially connected mCP-nets whose constituents are standard CP-nets. Interestingly, all these problems actually belong to various levels of the polynomial hierarchy, and some of them even belong to PTIME or LOGSPACE. Furthermore, for most of these problems, we provide completeness results, which show tight lower bounds for problems that (up to date) did not have any explicit non-obvious lower bound.