Many unique attributes of quantum cryptography arise from the no-cloning property of quantum information. We study this using two closely-related types of uncloneability game: no-cloning and monogamy-of-entanglement games. In a no-cloning game, a referee sends a quantum state encoding classical information to two cooperating players who split the state, then try simultaneously guessing the information, provided the key. In a monogamy-of-entanglement game, two cooperating players try to guess the referee's measurement result on a tripartite state the players prepared. In this work, we prove winning probability bounds on no-cloning games based on coset states, which have the interesting property that the players guess two different strings. We also show a rigidity property for the original monogamy-of-entanglement game, letting it be used as a test of separability. Finally, we apply these properties to construct a variety of novel cryptographic protocols for uncloneable encryption, quantum key distribution, bit commitment, and randomness expansion.