Recent experiments demonstrated quantum computational advantage in random circuit sampling and Gaussian boson sampling. However, it is unclear whether these experiments can lead to practical applications even after considerable research effort. On the other hand, simulating the quantum coherent dynamics of interacting spins has been considered as a potential first useful application of quantum computers, providing a possible quantum advantage. Despite evidence that simulating the dynamics of hundreds of interacting spins is challenging for classical computers, concrete proof is yet to emerge. We address this problem by proving that sampling from the output distribution generated by a wide class of quantum spin Hamiltonians is a hard problem for classical computers. Our proof is based on the Taylor series of the output probability, which contains the permanent of a matrix as a coefficient when bipartite spin interactions are considered. We devise a classical algorithm that extracts the coefficient using an oracle estimating the output probability. Since calculating the permanent is #P-hard, such an oracle does not exist unless the polynomial hierarchy collapses. With an anticoncentration conjecture, the hardness of the sampling task is also proven. Based on our proof, we estimate that an instance involving about 200 spins will be challenging for classical devices but feasible for intermediate-scale quantum computers with fault-tolerant qubits.