The echo state property, which is related to the dynamics of a neural network excited by input driving signals, is one of the well-known fundamental properties of recurrent neural networks. During the echo state, the neural network reveals an internal memory function that enables it to remember past inputs. Due to the echo state property, the neural network will asymptotically update its condition from the initial condition and is expected to exhibit temporally nonlinear input/output. As a physical neural network, we fabricated a quantum-dot network that is driven by sequential optical-pulse inputs and reveals corresponding outputs, by random dispersion of quantum-dots as its components. In the network, the localized optical energy of excited quantum-dots is allowed to transfer to neighboring quantum-dots, and its stagnation time due to multi-step transfers corresponds to the hold time of the echo state of the network. From the experimental results of photon counting of the fluorescence outputs, we observed nonlinear optical input/output of the quantum-dot network due to its echo state property. Its nonlinearity was quantitatively verified by a correlation analysis. As a result, the relation between the nonlinear input/outputs and the individual compositions of the quantum-dot network was clarified.