We measure the Hα and [O III] emission line properties as well as specific star formation rates (sSFRs) of spectroscopically confirmed 3 <z <6 galaxies in COSMOS from their observed colors versus redshift evolution. Our model describes consistently the ensemble of galaxies including intrinsic properties (age, metallicity, star formation history), dust attenuation, and optical emission lines. We forward-model the measured Hα equivalent widths (EW) to obtain the sSFR out to z ∼ 6 without stellar mass fitting. We find a strongly increasing rest-frame Hα EW that is flattening off above z ∼ 2.5 with average EWs of 300-600 Å at z ∼ 6. The sSFR is increasing proportionally to {(1+z)}2.4 at z <2.2 and to {(1+z)}1.5 at higher redshifts, indicative of a fast build-up of mass in high-z galaxies within e-folding times of 100-200 Myr at z ∼ 6. The redshift evolution at z > 3 cannot be fully explained in a picture of growth driven by cold accretion. We find a progressively increasing [O III]λ5007/Hβ ratio out to z ∼ 6, consistent with the ratios in local galaxies selected by increasing Hα EW (I.e., sSFR). This demonstrates the potential of using “local high-z analogs” to investigate the spectroscopic properties and relations of galaxies in the re-ionization epoch.