  Recently, semi-device independent protocols have attracted increasing
attention, guaranteeing security with few hypotheses and experimental
simplicity. In this paper, we demonstrate a many-outcomes scheme with the
binary phase-shift keying (BPSK) for a semi-device independent protocol based
on the energy assumption. We show in theory that the number of certified random
bits of the d-outcomes system outperforms the standard scheme
(binary-outcomes). Furthermore, we compare the results of two well-known
measurement schemes, homodyne and heterodyne detection. Lastly, taking into
account the experimental imperfections, we discuss the experimental feasibility
of the d-outcome design.
