  This paper presents a theoretical evaluation of coupled-spin entanglement in
the two-proton ($2p$) radioactive emission. The three-body model of $^{6}$Be
with the proton-proton interaction, which is adjusted to reproduce the
experimental energy release, is utilized. Time-dependent calculation is
performed to compute the coupled-spin state of the emitted two protons. The
spin-correlation function $S$ as the Clauser-Horne-Shimony-Holt (CHSH)
indicator is evaluated as $\abs{S} \cong 2.65$. Namely, the $2p$-spin
entanglement beyond the limit of local-hidden-variable (LHV) theory is
suggested. This entanglement is sensitive to the proton-proton interaction. The
short-lived (broad-width) $2p$~state has the weaker spin entanglement. In
parallel, the core-proton interactions do not harm this entanglement during the
time-dependent decaying process. The CHSH measurement can be a novel probe into
the effective nuclear interaction inside finite systems.
