Explore the intersection of quantum cryptography and computational complexity theory in this 59-minute lecture from the Simons Institute's Summer Cluster on Quantum Computing. Delve into how quantum mechanical principles can be leveraged to create cryptographic protocols and examine the computational complexity implications of quantum systems. Learn about quantum key distribution, quantum-resistant cryptographic schemes, and the theoretical foundations that underpin quantum cryptographic security. Investigate complexity classes specific to quantum computation and understand how quantum algorithms impact traditional cryptographic assumptions. Gain insights into current research directions where quantum information theory meets computational complexity, including discussions of quantum advantage in cryptographic applications and the challenges posed by quantum computers to classical cryptographic systems.