Population protocols are a model of computation in which an arbitrary number of indistinguishable finite-state agents interact in pairs. The goal of the agents is to decide by stable consensus whether their initial global configuration satisfies a given property, specified as a predicate on the set of configurations. The state complexity of a predicate is the number of states of a smallest protocol that computes it. Previous work by Blondin \textit{et al.} has shown that the counting predicates $x \geq η$ have state complexity $\mathcal{O}(\log η)$ for leaderless protocols and $\mathcal{O}(\log \log η)$ for protocols with leaders. We obtain the first non-trivial lower bounds: the state complexity of $x \geq η$ is $Ω(\log\log η)$ for leaderless protocols, and the inverse of a non-elementary function for protocols with leaders.