We consider the problem of measuring the spread of quantum states and justify the use of the Krylov basis to measure the spread of quantum states as a Hamiltonian system evolves, resulting in a measure of "spread complexity" akin to Krylov operator complexity but applied to states. In this basis, Hamiltonians become tridiagonal. The time-evolved thermofield double states in chaotic systems shows four regimes of this spread complexity: a linear ramp up to a peak that is exponential in the entropy, followed by a slope down to a plateau. We then derive analytical approximations for the mean of the tridiagonal matrix coefficients from the density of states in an general quantum system, as well as for both the mean and covariance for random matrices with a single-trace potential. Other quantities in the long time limit are shown to be computable as well. This is then applied in two ways: We show in examples that the covariance of the tridiagonal coefficients distinguish integrable and chaotic systems. Essentially, we provide a new set of tools capable of probing the detailed spectral statistics of quantum systems. We also find an application of the mean to infer the late time behavior of double scaled SYK. Lastly, as a somewhat separate topic, we shine a spotlight on the differences between the properties of eigenbasis chaos (e.g. the Eignestate Themalization Hypothesis) and spectral chaos (e.g. the eigenvalue repulsion found in chaotic systems). We show that without any constraints on the system or with a exponentially small tolerance in the constraints, these two properties are unrelated. We then provide a numerically-constructed example that shows that even with exact k-locality it may be possible to construct systems where the two properties do not agree. This suggests that careful considerations of typicality, restrictions, and/or the order of limits, may be needed to fully connect eigenbasis and spectral chaos.