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We study Hilbert-space localization of the many-body dynamics due to ergodicity breaking and analyze this effect in terms of the entanglement entropy and the entanglement spectrum. We find a transition from a regime driven by quantum tunneling to a regime that is dominated by boson-boson interaction, where the latter exhibits ergodicity breaking. Properties of this transition are captured by observation time averaging, which effectively suppresses the large dynamical entanglement fluctuations near the critical point. We employ this approach to the experimentally available Bosonic Josephson Junction. In this example the transition from a tunneling regime to Hilbert-space localization reveals clear signatures in the entanglement entropy and entanglement spectrum. Interestingly, the transition point is reduced by quantum effects in comparison to the well-known result of the mean-field approximation in the form of self-trapping. This indicates that quantum fluctuations reduce the classical self-trapping. Different scaling with the respect to the number of bosons $N$ are found in the tunneling and the localization regime: While the entanglement entropy grows {\it logarithmically} with $N$ in the tunneling regime, it increases {\it linearly} in the localized regime. Our results indicate that entanglement provides a concept for a sensitive diagnosis for the transition from a quantum tunneling regime to Hilbert-space localization.