This paper addresses the origin of the far-infrared (FIR) continuum of QSOs, based on the Quasar and ULIRG Evolution Study (QUEST) of nearby QSOs and ULIRGs using observations with the Spitzer Space Telescope. For 27 Palomar-Green QSOs at z <~ 0.3, we derive luminosities of diagnostic lines ([NeII]12.8um, [NeV]14.3um, [OIV]25.9um) and emission features (PAH7.7um emission which is related to star formation), as well as continuum luminosities over a range of mid- to far-infrared wavelengths between 6 and 60um. We detect star-formation related PAH emission in 11/26 QSOs and fine-structure line emission in all of them, often in multiple lines. The detection of PAHs in the average spectrum of sources which lack individual PAH detections provides further evidence for the widespread presence of PAHs in QSOs. Similar PAH/FIR and [NeII]/FIR ratios are found in QSOs and in starburst-dominated ULIRGs and lower luminosity starbursts. We conclude that the typical QSO in our sample has at least 30% but likely most of the far-infrared luminosity (~ 10^(10...12)Lsun) arising from star formation, with a tendency for larger star formation contribution at the largest FIR luminosities. In the QSO sample, we find correlations between most of the quantities studied including combinations of AGN tracers and starburst tracers. The common scaling of AGN and starburst luminosities (and fluxes) is evidence for a starburst-AGN connection in luminous AGN. Strong correlations of far-infrared continuum and starburst related quantities (PAH, low excitation [NeII]) offer additional support for the starburst origin of far-infrared emission.
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