We investigate the intra-cluster light (ICL) in the 6 Hubble Frontier Field clusters at $0.3<z<0.6$. We employ a new method, which is free from any functional form of the ICL profile, and exploit the unprecedented depth of this Hubble Space Telescope imaging to map the ICLs diffuse light out to clustrocentric radii $Rsim300$kpc ($mu_{rm ICL}sim27$mag arcsec$^{-2}$). From these maps, we construct radial color and stellar mass profiles via SED fitting and find clear negative color gradients in all systems with increasing distance from the Brightest Cluster Galaxy (BCG). While this implies older/more metal rich stellar components in the inner part of the ICL, we find the ICL mostly consists of a $<2$Gyr population, and plausibly originated with $log M_*/M_odot<10$ cluster galaxies. Further, we find 10-15% of the ICLs mass at large radii ($>150$kpc) lies in a younger/bluer stellar population ($sim1$Gyr), a phenomenon not seen in local samples. We attribute this light to the higher fraction of starforming/(post-)starburst galaxies in clusters at $zsim0.5$. Ultimately, we find the ICLs total mass to be $log M_{rm *}^{rm ICL}/M_odotsim11$-12, constituting 5%-20% of the clusters total stellar mass, or about a half of the value at $zsim0$. The above implies distinct formation histories for the ICL and BCGs/other massive cluster galaxies; i.e. the ICL at this epoch is still being constructed rapidly ($sim40M_odot$yr$^{-1}$), while the BCGs have mostly completed their evolution. To be consistent with the ICL measurements of local massive clusters, such as the Virgo, our data suggest mass acquisition mainly from quiescent cluster galaxies is the principal source of ICL material in the subsequent $sim$5 Gyr of cosmic time.
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