In the worst case, the computational effort to evaluate a QPLEX map grows at least linearly in the cardinality of the support of the marginal pmf of the counter vector. Thus, for models involving many counters, such as (large) network models, evaluating the QPLEX map (and calculating QPLEX iterates) can require an effort that is exponential in the number of counters, resulting in a second curse of dimensionality. We call this the curse of dimensionality for counters. Instead of working with pmfs of a counter vector and a label vector, we overcome this curse of dimensionality for counters in Part II by working with collections of pmfs of (generally) fewer variables.