We resolve the computational complexity of three problems known as Necklace Splitting, Consensus-Halving, and Discrete Ham sandwich, showing that they are PPAcomplete. For NECKLACE SPLITTING, this result is specific to the important special case in which two thieves share the necklace. These are the first PPA-completeness results for problems whose definition does not contain an explicit circuit, thus settling the status of PPA as a class that captures the complexity of such "natural' problems.