For every $k \ge 1$, alternating machine acceptance with $k$ blocks is $\Sigma_k^p$-complete when the first block is existential and $\Pi_k^p$-complete when it is universal, under first-order reductions; and a decision problem is in $\Sigma_k^p$, respectively $\Pi_k^p$, if and only if it reduces to the corresponding acceptance problem by an ordered first-order reduction. Each level of the logically defined hierarchy is thus the corresponding level of the alternating-machine hierarchy of Chandra, Kozen and Stockmeyer; at one block these are the nondeterministic machine and its dual, the machine model of coNP. Membership reads a run as a game of $k$ rounds, each guessing one walk; hardness builds, inside the instance, the machine of a quantified Boolean formula, which sweeps the tape once per quantifier block.