Model checking of asynchronous systems is traditionally based on the interleaving model, where an execution is modeled by a total order between events. Recently, the use of partial order semantics that allows independent events of concurrent processes to be unordered is becoming popular. Temporal logics that are interpreted over partial orders allow specifications relating global snapshots, and permit reduction algorithms to generate only one representative linearization of every possible partial-order execution during state-space search. This paper considers the satisfiability and the model checking problems for temporal logics interpreted over partially ordered sets of global configurations. For such logics, only undecidability results have been proved previously. In this paper, we present an EXPSPACE decision procedure for a fragment that contains an eventuality operator and its dual. We also sharpen previous undecidability results, which used global predicates over configurations. We show that although our logic allows only local propositions (over events), it becomes undecidable when adding some natural until operator.