Massive quiescent compact galaxies have been discovered at high redshifts, associated with rapid compaction and cessation of star formation (SF). In this work we set out to quantify the time-scales in which SF is quenched in compact galaxies at intermediate redshifts. For this, we select a sample of green valley galaxies within the COSMOS field in the midst of quenching their SF at $0.5<z<1.0$ that exhibit varying degrees of compactness. Based on the H$delta$ absorption line and the 4000 AA break of coadded zCOSMOS spectra for sub-samples of normal-sized and compact galaxies we determine quenching time-scales as a function of compactness. We find that the SF quenching time-scales in green valley compact galaxies are much shorter than in normal-sized ones. In an effort to understand this trend, we use the Illustris simulation to trace the evolution of the SF history, the growth rate of the central super massive black hole (SMBH) {bf and the AGN-feedback in compact and normal-sized galaxies. We find that the key difference between their SF quenching time-scales is linked to the mode of the AGN-feedback. In the compact galaxies predominates the kinematic-mode, which is highly efficient at quenching the SF by depleting the internal gas. On the normal-sized galaxies, the prevailing thermal-mode injects energy in the circumgalactic gas, impeding the cold gas supply and quenching the SF via the slower strangulation mechanism.} These results are consistent with the violent disk instability and gas-rich mergers scenarios, followed by strong AGN and stellar feedback. Although this kind of event is most expected to occur at $z=2-3$, we find evidences that the formation of compact quiescent galaxies can occur at $z<1$.
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