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We report CO(5$\rightarrow$4) and CO(6$\rightarrow$5) line observations inthe dusty starbursting galaxy CRLE ($z = 5.667$) and the main-sequence (MS)galaxy HZ10 ($z = 5.654$) with the Northern Extended Millimeter Array (NOEMA).CRLE is the most luminous $z>5$ starburst in the COSMOS field and HZ10 is themost gas-rich "normal" galaxy currently known at $z>5$. We find lineluminosities for CO(5$\rightarrow$4) and CO(6$\rightarrow$5) of (4.9 $\pm$ 0.5)and (3.8 $\pm$ 0.4) $\times$ 10$^{10}$ K km s$^{-1}$ pc$^{2}$ for CRLE andupper limits of $< 0.76$ and $< 0.60$ $\times$ 10$^{10}$ K km s$^{-1}$ pc$^{2}$for HZ10, respectively. The CO excitation of CRLE appears comparable to other$z>5$ dusty star-forming galaxies (DSFGs). For HZ10, these line luminositylimits provide the first significant constraints of this kind for a MS galaxyat $z > 5$. We find the upper limit of $L'_{5\rightarrow4}/L'_{2\rightarrow1}$in HZ10 could be similar to the average value for MS galaxies around $z\approx1.5$, suggesting that MS galaxies with comparable gas excitation may alreadyhave existed one billion years after the Big Bang. For CRLE we determine themost likely values for the H$_2$ density, kinetic temperature and dusttemperature based on excitation modeling of the CO line ladder. We also derivea total gas mass of $(7.1 \pm 1.3) \times 10^{10} M_\odot$. Our findingsprovide some of the currently most detailed constraints on the gas excitationthat sets the conditions for star formation in a galaxy protoclusterenvironment at $z > 5$.

Molecular Gas Excitation of the Massive Dusty Starburst CRLE and the Main-Sequence Galaxy HZ10 at z=5.7 in the COSMOS Field