We elucidate physical aspects of path signatures by formulating randomised path developments within the framework of matrix models in quantum field theory. Using tools from physics, we introduce a new family of randomised path developments and derive corresponding loop equations. We then interpret unitary randomised path developments as time evolution operators on a Hilbert space of qubits. This leads to a definition of a quantum path signature feature map and associated quantum signature kernel through a quantum circuit construction. In the case of the Gaussian matrix model, we study a random ensemble of Pauli strings and formulate a quantum algorithm to compute such kernel.