  We generalize the hidden-fermion family of neural network quantum states to
encompass both continuous and discrete degrees of freedom and solve the nuclear
many-body Schr\"odinger equation in a systematically improvable fashion. We
demonstrate that adding hidden nucleons to the original Hilbert space
considerably augments the expressivity of the neural-network architecture
compared to the Slater-Jastrow ansatz. The benefits of explicitly encoding in
the wave function point symmetries such as parity and time-reversal are also
discussed. Leveraging on improved optimization methods and sampling techniques,
the hidden-nucleon ansatz achieves an accuracy comparable to the
numerically-exact hyperspherical harmonic method in light nuclei and to the
auxiliary field diffusion Monte Carlo in $^{16}$O. Thanks to its polynomial
scaling with the number of nucleons, this method opens the way to
highly-accurate quantum Monte Carlo studies of medium-mass nuclei.
