  We present correlations between 9 CO transition ($J=4-3$ to $12-11$) and
beam-matched far-infrared (Far-IR) luminosities ($L_{\mathrm{FIR},\,b}$) among
167 local galaxies, using {\it{Herschel}} Spectral and Photometric Imaging
Receiver Fourier Transform Spectrometer (SPIRE; FTS) spectroscopic data and
Photoconductor Array Camera and Spectrometer (PACS) photometry data. We adopt
entire-galaxy FIR luminosities ($L_{\mathrm{FIR},\,e}$) from the {\it{IRAS}}
Revised Bright Galaxy Sample and correct to $L_{\mathrm{FIR},\,b}$ using PACS
images to match the varying FTS beam sizes. All 9 correlations between
$L'_{\mathrm{CO}}$ and $L_{\mathrm{FIR},\,b}$ are essentially linear and tight
($\sigma$=0.2-0.3 dex dispersion), even for the highest transition, $J=12-11$.
This supports the notion that the star formation rate (SFR) is linearly
correlated with the dense molecular gas
($n_{\mathrm{H}_2}\gtrsim10^{4-6}\,cm^{-3}$). We divide the entire sample into
three subsamples and find that smaller sample sizes can induce large
differences in the correlation slopes. We also derive an average CO spectral
line energy distribution (SLED) for the entire sample and discuss the implied
average molecular gas properties for these local galaxies. We further extend
our sample to high-{\it{z}} galaxies with CO $J=5-4$ data from the literature
as an example, including submillimeter galaxies (SMGs) and "normal"
star-forming BzKs. BzKs have similar FIR/CO(5-4) ratios as that of local
galaxies, an follow well the locally-determined correlation, whereas SMG ratios
fall around or slightly above the local correlation with large uncertainties.
Finally, by including Galactic CO($J=10-9$) data as well as very limited
high-{\it{z}} CO $J=10-9$ data, we verify that the CO(10-9) -- FIR correlation
successfully extends to Galactic young stellar objects, suggesting that linear
correlations are valid over 15 orders of magnitude.
