Quantum computing promises to speed up machine-learning algorithms. However, noisy intermediate-scale quantum (NISQ) devices pose engineering challenges to realizing quantum machine-learning (QML) advantages. Recently, a series of QML computational models inspired by the noise-tolerant dynamics of the brain has emerged as a means to circumvent the hardware limitations of NISQ devices. In this paper, we introduce a quantum version of a recurrent neural network (RNN), a well-known model for neural circuits in the brain. Our quantum RNN (qRNN) makes use of the natural Hamiltonian dynamics of an ensemble of interacting spin-1/2 particles as a means for computation. In the limit where the Hamiltonian is diagonal, the qRNN recovers the dynamics of the classical version. Beyond this limit, we observe that the quantum dynamics of the qRNN provide it with quantum computational features that can aid it in computation. To this end, we study a fixed-geometry qRNN, i.e., a quantum reservoir computer, based on arrays of Rydberg atoms and show that the Rydberg reservoir is indeed capable of replicating the learning of several cognitive tasks such as multitasking, decision making, and long-term memory by taking advantage of several key features of this platform such as interatomic species interactions and quantum many-body scars.