The paper extends and generalizes the notion of synchronized alternation, studied in the literature. We show that, though the underlying machine for the synchronized alternation is alternating, the nature of message broadcasting is nondeterministic. Therefore, we introduce a new computational model, the so-called communicating alternation machines; alternating machines equipped with an alternating communication. Then we show that the two-level communication, i.e., existential and universal, can be generalized uniformly arbitrarily many times, giving communication with r communication levels. This allows, among others, to extend the well-known characterization of DLOGSPACE, NLOGSPACE, P, and PSPACE by deterministic, nondeterministic, alternating, and synchronized alternating two-way read-only multihead finite automata. The above characterization represents just the first four members of an infinite hierarchy of multihead automata, representing the entire fundamental complexity hierarchy. The first “new” class added is DEXPTIME, corresponding to the two-level communicating alternation automata.