We investigate the frontier between classical and quantum plasmonics in highly doped semiconductor layers. The choice of a semiconductor platform instead of metals for our study permits an accurate description of the quantum nature of the electrons constituting the plasmonic response, which is a crucial requirement for quantum plasmonics. Our quantum model allows us to calculate the collective plasmonic resonances from the electronic states determined by an arbitrary one-dimensional potential. Our approach is corroborated with experimental spectra, realized on a single quantum well, in which higher order longitudinal plasmonic modes are present. We demonstrate that their energy depends on the plasma energy, as is also the case for metals, but also on the size confinement of the constituent electrons. This work opens the way toward the applicability of quantum engineering techniques for semiconductor plasmonics.