Lecture I: Hilbert Space is Huge 1 Preface 2 How Huge? 3 Volume of CP(N) 4 Relative Complexity 5 Dual Role of Unitaries 6 Volume of SU(2K) 7 Exploring SU(2K) 7.1 Relative Complexity of Unitaries 7.2 Complexity is Discontinuous 8 Graph Theory Perspective 8.1 Collisions and Loops 9 The Second Law of Quantum Complexity 9.1 Hamiltonian Evolution II Lecture II: Black Holes and the Second Law of Complexity 10 Preface 11 The Black Hole-Quantum Circuit Correspondence 11.1 Two Problems 11.2 Circuits and Black Holes 12 The Growth of Wormholes 12.1 Properties of Growth 12.

2 Rindler Time and CV 13 Exponential Time Breakdown of GR 13.1 C=V 14 Precursors 14.1 The Epidemic Model 14.2 Lyapunov and Rindler 15 Precursors and Black Holes 15.1 Instability of White Holes 16 Complexity and Firewalls 16.1 Firewalls are Fragile 16.2 What Happens After Exponential Time? 16.3 The Fragility of Complexity Equilibrium 17 Do Typical States have Firewalls? 17.

1 AdS Black Holes 17.2 Evaporating Black Holes Lecture III: The Thermodynamics of Complexity 18 Preface 19 Negentropy 20 Uncomplexity 20.1 The Auxiliary System 20.2 Combining Auxiliary Systems 21 Uncomplexity as a Resource 22 The Power of One Clean Qubit 22.1 The Protocol 22.2 Expending Uncomplexity and Negentropy 23 Spacetime and Uncomplexity 23.1 CA 23.2 Geometric Interpretation of Uncomplexity Conclusion.