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As NASA spacecraft explore deeper into the cosmos, speed-of-light-limited signal delays make it increasingly impractical to command missions from Earth. Future spacecraft will need greater onboard computing capacity to mimic human-level intelligence and autonomy. The solution might come from quantum computers, which offer properties of size, power, and robustness that are ideally suited to the space environment. The potential of quantum technologies goes far beyond enhanced computing capacity. Future space missions will involve direct participation of non-NASA scientists. This will necessitate allowing more open access to spacecraft systems via free-space communication links. Quantum cryptography would allow such channels to be made absolutely secure and invulnerable to attack by malevolent hackers. To explore these possibilities, this article describes the progress to date in understanding how quantum computers and related quantum information-processing devices might advance space exploration.