This paper investigates the relationship between Krylov complexity (C_K) and Spread complexity (C_S) for density matrix operators, focusing on their behavior in various quantum systems. The study analyzes both analytical and numerical examples across different Hilbert spaces, including two-dimensional systems, qubit states, quantum harmonic oscillators, and random matrix theories. Key findings include a correspondence between moment-generating functions of Lanczos coefficients and survival amplitudes, and an early-time equivalence between C_K and 2C_S for generic pure states. For maximally entangled pure states, the moment-generating function of C_K becomes the Spectral Form Factor, and at late times, C_K is related to NC_S for N ≥ 2. The paper also explores the connection between Krylov complexity and Spread complexity in the context of the thermofield double (TFD) state, showing that C_K = 2C_S holds for all times when N = 2. The study highlights subtleties in the averaging approach and provides insights into the behavior of complexities in random matrix theories and quantum systems. The results demonstrate that Krylov complexity can be related to Spread complexity in various scenarios, offering a deeper understanding of quantum complexity measures.This paper investigates the relationship between Krylov complexity (C_K) and Spread complexity (C_S) for density matrix operators, focusing on their behavior in various quantum systems. The study analyzes both analytical and numerical examples across different Hilbert spaces, including two-dimensional systems, qubit states, quantum harmonic oscillators, and random matrix theories. Key findings include a correspondence between moment-generating functions of Lanczos coefficients and survival amplitudes, and an early-time equivalence between C_K and 2C_S for generic pure states. For maximally entangled pure states, the moment-generating function of C_K becomes the Spectral Form Factor, and at late times, C_K is related to NC_S for N ≥ 2. The paper also explores the connection between Krylov complexity and Spread complexity in the context of the thermofield double (TFD) state, showing that C_K = 2C_S holds for all times when N = 2. The study highlights subtleties in the averaging approach and provides insights into the behavior of complexities in random matrix theories and quantum systems. The results demonstrate that Krylov complexity can be related to Spread complexity in various scenarios, offering a deeper understanding of quantum complexity measures.