  We analyze the stellar populations of a sample of 62 massive (log Mstar/Msun
> 10.7) galaxies in the redshift range 1 < z < 1.6, with the main goal of
investigating the role of recent quenching in the size growth of quiescent
galaxies. We demonstrate that our sample is not biased toward bright, compact,
or young galaxies, and thus is representative of the overall quiescent
population. Our high signal-to-noise ratio Keck LRIS spectra probe the
rest-frame Balmer break region which contains important absorption line
diagnostics of recent star formation activity. We obtain improved measures of
the various stellar population parameters, including the star-formation
timescale tau, age and dust extinction, by fitting templates jointly to both
our spectroscopic and broad-band photometric data. We identify which quiescent
galaxies were recently quenched and backtrack their individual evolving
trajectories on the UVJ color-color plane finding evidence for two distinct
quenching routes. By using sizes measured in the previous paper of this series,
we confirm that the largest galaxies are indeed among the youngest at a given
redshift. This is consistent with some contribution to the apparent growth from
recent arrivals, an effect often called progenitor bias. However, we calculate
that recently-quenched objects can only be responsible for about half the
increase in average size of quiescent galaxies over a 1.5 Gyr period,
corresponding to the redshift interval 1.25 < z < 2. The remainder of the
observed size evolution arises from a genuine growth of long-standing quiescent
galaxies.
