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As post-quantum cryptography (PQC) nears standardization and eventual deployment, it is increasingly important to understand the security of the implementations of selected schemes. In this paper, we conduct such an investigation, uncovering concerning findings about many of the finalists of the NIST PQC standardization competition. Specifically, we show Rowhammer-based attacks on the Kyber and BIKE Key Exchange Mechanisms and the Dilithium Digital Signature scheme that enable complete recovery of the secret key with only a moderate amount of effort – no supercomputers, or months of precomputation. Moreover, we experimentally carry out our attacks using a combination of Rowhammer, performance degradation, and memory massaging techniques, showing that our attacks are practically feasible. Our results show that such side-channel based attacks are a critical concern and need to be considered when new cryptographic schemes are standardized, when standard implementations are developed, and when instances are deployed. We conclude with recommendations on implementation techniques that harden cryptographic schemes against Rowhammer attacks.