  We reanalyze the mid-infrared (5-40 $\mu$m) Spitzer spectra of 86
low-redshift ($z < 0.5$) Palomar-Green quasars to investigate the nature of
polycyclic aromatic hydrocarbon (PAH) emission and its utility as a star
formation rate (SFR) indicator for the host galaxies of luminous active
galactic nuclei (AGNs). We decompose the spectra with our recently developed
template-fitting technique to measure PAH fluxes and upper limits, which we
interpret using mock spectra that simulate the effects of AGN dilution. While
luminous quasars can severely dilute and affect the detectability of emission
lines, PAHs are intrinsically weak in some sources that are otherwise gas-rich
and vigorously forming stars, conclusively demonstrating that powerful AGNs
destroy PAH molecules. Comparing PAH-based SFRs with independent SFRs derived
from the mid-infrared fine-structure neon lines and the total infrared
luminosity reveals that PAHs can trace star formation activity in quasars with
bolometric luminosities $\lesssim 10^{46}\, \rm erg\,s^{-1}$, but increasingly
underestimate the SFR for more powerful quasars, typically by $\sim 0.5$ dex.
Relative to star-forming galaxies and low-luminosity AGNs, quasars have a
comparable PAH 11.3 $\mu$m/7.7 $\mu$m ratio but characteristically lower ratios
of 6.2 $\mu$m/7.7 $\mu$m, 8.6 $\mu$m/7.7 $\mu$m, and 11.3 $\mu$m/17.0 $\mu$m.
We suggest that these trends indicate that powerful AGNs preferentially destroy
small grains and enhance the PAH ionization fraction.
