Recent investigations have provided evidences that many massive (M_* > 10^{11} M_&sun;) spheroid galaxies in the early universe (z ≥ 1.5) were extremely compact (r_e ˜ 1 kpc). According to some galaxy formation theories, a fraction of these massive galaxies in the local Universe would represent surviving relics of the high-z massive galaxies. We present a detailed study of the kinematics and the stellar populations for seven local compact galaxiesfor which we obtained high-quality long-slit spectra. We find that thesegalaxies show metallicity values around solar. However, unlike common, average-sized localellipticals of similar velocity dispersion, they show young (<2 Gyr) meanluminosity weighted ages as well as a clearly different abundance ratio pattern.In addition, almost all these massive compact local galaxies show strong rotation curves.The stellar population findings point that these local objects cannot be the relics ofhigh-z compact ellipticals. The most stricking result we have found is that our fullspectrum-fitting analysis shows that the stellar populations of our galaxies are mainly contributed by young populations.