Simulating quantum many‐body dynamics on classical computers is a challenging problem due to the exponential growth of the Hilbert space. Artificial neural networks have recently been introduced as a new tool to approximate quantum many‐body states. The variational power of the restricted Boltzmann machine quantum states and different shallow and deep neural autoregressive quantum states to simulate the global quench dynamics of a non‐integrable quantum Ising chain is benchmarked. It is found that the number of parameters required to represent the quantum state at a given accuracy increases exponentially in time. The growth rate is only slightly affected by the network architecture over a wide range of different design choices: shallow and deep networks, small and large filter sizes, dilated and normal convolutions, and with and without shortcut connections.