Several input read-modes for alternating Turing machines have been proposed in the literature. For each input read-mode and for each fixed integer k ≥ 1, a precise circuit characterization is established for log-time alternating Turing machines of k alternations, which is a nontrivial refinement of Ruzzo's circuit characterization of alternating Turing machines. Complete languages in strong sense for each level of the log-time hierarchy are presented, refining a result by Buss. The class GC(s(n), Π_k^B) is investigated, which is the class of languages accepted by log-time alternating Turing machines of k alternations enhanced by an extra ability of guessing a string of length s(n). A systematic technique is developed to show that for many functions s(n) and for every integer k>1, the class GC(s(n), Π_k^B) has natural complete languages. Connections of these results to computational optimization problems are exhibited.