  We report $J^\pi = 0^+$ ground-state energies and point-proton radii of
$^4$He, $^8$Be, $^{12}$C, $^{16}$O and $^{20}$Ne nuclei calculated by the $ab$
$initio$ no-core Monte Carlo shell model with the JISP16 and Daejeon16 nonlocal
$NN$ interactions. Ground-state energies are obtained in the basis spaces up to
seven oscillator shells ($N_{\rm shell} = 7$) with several oscillator energies
($\hbar \omega$) around the optimal oscillator energy for the convergence of
ground-state energies. These energy eigenvalues are extrapolated to obtain
estimates of converged ground state energies in each basis space using energy
variances of computed energy eigenvalues. We further extrapolate these
energy-variance-extrapolated energies obtained in the finite basis spaces to
infinite basis-space results with an empirical exponential form. This form
features a dependence on the basis-space size but is independent of the value
of $\hbar\omega$ used for the harmonic-oscillator basis functions. Point-proton
radii for these states of atomic nuclei are also calculated following
techniques employed for the energies. From these results, it is found that the
Daejeon16 $NN$ interaction provides good agreement with experimental data up to
approximately $^{16}$O, while the JISP16 $NN$ interaction provides good
agreement with experimental data up to approximately $^{12}$C. Beyond these
nuclei, the interactions produce overbinding accompanied by radii that are too
small. These findings suggest and encourage further revisions of nonlocal $NN$
interactions towards the investigation of nuclear structure in heavier-mass
regions.
