The abundance of data to be processed calls for new computing paradigms, which could accommodate, and directly map artificial neural network architectures at the hardware level. Neuromorphic computing has emerged as a potential solution, proposing the implementation of artificial neurons and synapses on physical substrates. Conventionally, neuromorphic platforms are deployed in complementary metal-oxide-semiconductor technology. However, such implementations still cannot compete with the highly energy-efficient performance of the brain. This calls for novel ultra-low-power nano-scale devices with the possibility of upscaling for the implementation of complex networks. In this paper, a multi-state spin− orbit torque (SOT) synapse based on the three-terminal perpendicular anisotropy magnetic tunnel junction (P-MTJ) is proposed. In this implementation, P-MTJs use common heavy metals but with different cross-section areas, thereby creating multiple states that can be harnessed to implement synapses. The proposed multi-state SOT synapse can solve the state-limited issue of spin-based synapses. Moreover, it is shown that the proposed multi-state SOT synapse can be programmed to reproduce the spike-timing-dependent plasticity learning algorithm.