Quantum complexity has been conjectured to be relevant for black hole interiors. In particular, Susskind has conjectured a generic form of complexity growth and saturation for maximally chaotic systems like black holes. Using geodesic complexity techniques, we study the complexity of time evolution in free, integrable, and chaotic systems with N degrees of freedom. We demonstrate that the growth rate of this complexity is related to more familiar properties of physical theories like 1) the existence of creation/annihilation operators and 2) smeared thermal two-point functions. Using these properties, we find that the complexity saturates in free, integrable, and chaotic theories at time scales which are on the order of N^{1/2}, polynomial in N, and exponential in N, respectively. Our results in the free theory improve an existing bound in the complexity theory literature, and our results in chaotic theories are based only on coarse properties of the spectrum and eigenstates.
