The goal of this work is to study the evolution of high-redshift (z≥ 1.4) quiescent galaxies over an effective area of ˜1.7 deg2 in the COSMOS field. Galaxies have been divided according to their star formation activity and the evolution of the different populations, in particular of the quiescent galaxies, has been investigated in detail. We have studied an IRAC (mag 3.6 μm < 22.0) selected sample of ˜18 000 galaxies at z≥ 1.4 in the COSMOS field with multiwavelength coverage extending from the U band to the Spitzer 24 μm one. We have derived accurate photometric redshifts (?) through a SED-fitting procedure. Other important physical parameters [masses, ages and star formation rates (SFR)] of the galaxies have been obtained using Maraston models. We have divided our sample into actively star-forming, intermediate and quiescent galaxies depending on their specific star formation rate (SSFR = SFR/M). We have computed the galaxy stellar mass function (GSMF) of the total sample and the different populations at z= 1.4-3.0. We have studied the properties of high-redshift quiescent galaxies finding that they are old (1-4 Gyr), massive (˜ 1010.65 M&sun;), weakly star-forming stellar populations with low dust extinction [E(B-V) ≤ 0.15] and small e-folding time-scales (τ˜ 0.1-0.3 Gyr). We observe a significant evolution of the quiescent stellar mass function from 2.5 < z < 3.0 to 1.4 < z < 1.6, increasing by ˜1 dex in this redshift interval. We find that z˜ 1.5 is an epoch of transition of the GSMF: while the GSMF at z≳ 1.5 is dominated by the star-forming galaxies at all stellar masses, at z≲ 1.5 the contribution to the total GSMF of the quiescent galaxies is significant and becomes higher than that of the star-forming population for M≥ 1010.75 M&sun;. The fraction of star-forming galaxies decreases from 60-20 per cent from z˜ 2.5-3.0 to 1.4-1.6 for M˜ 1011.0 M&sun;, while the quiescent population increases from 10-50 per cent at the same redshift and mass intervals. We compare the fraction of quiescent galaxies derived with that predicted by theoretical models and find that the Kitzbichler & White model, implemented on the Millennium Simulation, is the one that better reproduces the shape of the data. Finally, we calculate the stellar mass density of the star-forming and quiescent populations as a function of redshift and find that there is already a significant number of quiescent galaxies at z > 2.5 (logρ[M&sun; Mpc-3]˜ 6), meaning that efficient star formation had to take place before that time.