  Some high-z active galactic nuclei (AGNs) are found to reside in extreme
star-forming galaxies, such as hyper-luminous infrared galaxies (HyLIRGs), with
AGN-removed $L_{\rm{IR}}$ of $>10^{13} L_{\rm{\odot}}$. In this paper, we
report NOEMA observations of six apparent starburst HyLIRGs associated with
optical quasars at $z\sim2-3$ in the Stripe 82 field, to study their dust and
molecular CO properties. Five out of the six candidates are detected with
CO(4-3) or CO(5-4) emission, and four in 2mm dust continuum. Based on the
linewidth-$L'_{\rm{CO(1-0)}}$ diagnostics, we find that four galaxies are
likely unlensed or weakly lensed sources. The molecular gas mass is in the
range of $\mu M_{\rm{H_2}} \sim0.8-9.7\times10^{10} M_{\odot}$ (with $\alpha =
0.8 M_{\odot} (\rm{K km s^{-1} pc^2})^{-1}$ and $\mu$ is the unknown possible
gravitational magnification factor). We fit their SEDs, after including the
observed 2mm fluxes and upper limits, and estimate their apparent (uncorrected
for possible lensing effect) star formation rates ($\mu$SFRs) to be
$\sim400-2500$ $M_{\rm{\odot}} \rm{yr^{-1}}$ with depletion time of
$\sim20-110$ Myr. We notice interesting offsets, of $\sim10-40$ kpc spatially
or $\sim1000-2000$ km s$^{-1}$ spectroscopically, between the optical quasar
and the mm continuum or CO emissions. The observed velocity shift is likely
related to the blueshifted broad-emission-line region of quasars, though
mergers or recoiling black holes are also possible causes, which can explain
the spatial offset and the high intrinsic SFRs in the HyLIRG-quasar systems.
