One-counter processes are pushdown systems over a singleton stack alphabet (plus a stack-bottom symbol). We study the complexity of two closely related verification problems over one-counter processes: model checking with the temporal logic EF, where formulas are given as directed acyclic graphs, and weak bisimilarity checking against finite systems. We show that both problems are $\P^\NP$-complete. This is achieved by establishing a close correspondence with the membership problem for a natural fragment of Presburger Arithmetic, which we show to be$\P^\NP$-complete. This fragment is also a suitable representation for the global versions of the problems. We also show that there already exists a fixed EF formula(resp. a fixed finite system) such that model checking (resp. weak bisimulation) over one-counter processes is hard for $\P^{\NP[\log]}$. However, the complexity drops to $\P$ if the one-counter process is fixed.