Summary: We study the class of binary coded versions of unary languages that can be accepted by alternating machines with \(\log\log n\) space. We show that there exists a binary PSpace-complete language \(\mathcal {L}\) such that the unary coded version of \(\mathcal {L}\) is in \(\text{\textsc{ASpace}}(\log\log n)\). Consequently, the standard translation between unary languages accepted with \(\log\log n\) space and binary languages accepted with \(\log n\) space works for alternating machines if and only if \(\mathrm{P} = \text{\textsc{PSpace}}\). In general, if a binary language is accepted deterministically in \(2^n \cdot n^{O (1)}\) time and, simultaneously, in \(n^{O(1)}\) space – which covers many PSpace-complete problems – then its unary coded version is accepted by an alternating Turing machine using an initially delimited worktape of size \(\log\log n\). This unexpected power follows from the fact that, with an auxiliary worktape of size \(O(\log\log n)\) on a unary input \(1^n\), an alternating machine can simulate a stack with \(\log n\) bits, representing the contents of the stack by its input head position. The standard push/pop operations on the stack are implemented by moving the head along the input.