The notion of Online State Complexity, introduced by Karp in 1967, quantifies the amount of states required to solve a given problem using an online algorithm, which is represented by a deterministic machine scanning the input from left to right in one pass.

In this paper, we extend the setting to alternating machines as introduced by Chandra, Kozen and Stockmeyer in 1976: such machines run independent passes scanning the input from left to right and gather their answers through boolean combinations.

We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique.
The first is a hierarchy theorem, stating that the polynomial hierarchy of alternating online state complexity is infinite, and the second is a linear lower bound on the alternating online state complexity of the prime numbers written in binary.
This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.

The notion of online space complexity, introduced by Karp in 1967, quantifies the amount of states required to solve a given problem using an online algorithm,
represented by a machine which scans the input exactly once from left to right.
In this paper, we study alternating machines as introduced by Chandra, Kozen and Stockmeyer in 1976.

We devise a lower bound technique relying on boundedly generated lattices of languages, and give two applications of this technique.
The first is a hierarchy theorem for languages of polynomial alternating online space complexity,
and the second is a linear lower bound on the alternating online space complexity of the prime numbers written in binary.
This second result strengthens a result of Hartmanis and Shank from 1968, which implies an exponentially worse lower bound for the same model.