Explore the intricacies of random quantum circuits, phase transitions, and complexity in this 41-minute lecture by Aram Harrow at the Institute for Advanced Study. Delve into topics such as the complexity of random quantum circuits, random circuit sampling conjecture, and Haar-random unitaries. Examine the necessity of Haar randomness and investigate pseudo-random unitaries. Discover applications of designs and analyze 1-D random circuits depth T, as well as random circuits in higher dimensions. Study general geometries and the proof of D=1 result. Investigate low-depth circuits, tensor contraction in 1-D, and simulating 2-D circuits. Learn about cheaper tensor contraction and approximate simulation techniques. Evaluate the effectiveness of the algorithm and explore random tensor networks. Conclude with open questions in the field of quantum complexity.