We detected 10 compact galaxy groups (CGs) at $z=0$ in the semi-analytic galaxy catalog of Guo et al. (2011) for the milli-Millennium Cosmological Simulation (sCGs in mGuo2010a). We aimed to identify potential canonical pathways for compact group evolution and thus illuminate the history of observed nearby compact groups. By constructing merger trees for $z=0$ sCG galaxies, we studied the cosmological evolution of key properties, and compared them with $z=0$ Hickson CGs (HCGs). We found that, once sCG galaxies come within 1 (0.5) Mpc of their most massive galaxy, they remain within that distance until $z=0$, suggesting sCG birth redshifts. At $z=0$ stellar masses of sCG most-massive galaxies are within $10^{10} lesssim M_{ast}/M_{odot} lesssim 10^{11}$. In several cases, especially in the two 4- and 5-member systems, the amount of cold gas mass anti-correlates with stellar mass, which in turn correlates with hot gas mass. We define the angular difference between group members 3D velocity vectors, $Deltatheta_{rm vel}$, and note that many of the groups are long-lived because their small values of $Deltatheta_{rm vel}$ indicate a significant parallel component. For triplets in particular, $Deltatheta_{rm vel}$ values range between $20^{circ}$ and $40^{circ}$ so that galaxies are coming together along roughly parallel paths, and pairwise separations do not show large pronounced changes after close encounters. The best agreement between sCG and HCG physical properties is for $M_{ast}$ galaxy values, but HCG values are higher overall, including for SFRs. Unlike HCGs, due to a tail at low SFR and $M_{ast}$, and a lack of $M_{ast}gtrsim 10^{11}M_{odot}$ galaxies, only a few sCG galaxies are on the star-forming main sequence.
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