The review "Quantum Dynamics in Krylov Space: Methods and Applications" by Pratik Nandy et al. explores the use of Krylov subspace methods to efficiently describe quantum evolution, particularly in nonequilibrium many-body systems with large Hilbert spaces. The authors provide an updated overview of recent developments, focusing on the quantum evolution of operators in the Heisenberg picture and the quantification of operator growth through Krylov complexity. They discuss the universal operator growth hypothesis, its relation to quantum chaos and scrambling, and its implications for quantum field theory, holography, integrability, quantum control, and quantum computing. The review also covers the Lanczos algorithm, coherent states, operator size concentration, and Krylov entropy, highlighting the computational efficiency and theoretical insights provided by Krylov subspace methods.The review "Quantum Dynamics in Krylov Space: Methods and Applications" by Pratik Nandy et al. explores the use of Krylov subspace methods to efficiently describe quantum evolution, particularly in nonequilibrium many-body systems with large Hilbert spaces. The authors provide an updated overview of recent developments, focusing on the quantum evolution of operators in the Heisenberg picture and the quantification of operator growth through Krylov complexity. They discuss the universal operator growth hypothesis, its relation to quantum chaos and scrambling, and its implications for quantum field theory, holography, integrability, quantum control, and quantum computing. The review also covers the Lanczos algorithm, coherent states, operator size concentration, and Krylov entropy, highlighting the computational efficiency and theoretical insights provided by Krylov subspace methods.