The early civilizations of the Americas left many advances in Science that even we have yet to match. For example the Incas left us a calendar that was more accurate than our own, the real day has 23 hours and just under 59 minutes, our calendar rounds off to an even 24 hours, as a result we have and extra day every 4 years which is still off by some 100 seconds every year which continues to boggle our scientists as to how they accomplished an efficient calendar more accurate than our own without the advanced technology we have. This knowledge was important to the Incas so that they could plant and harvest at just the right moment for the most effiecient crop on their equally boggling Terrace Farmland.
However, by far the most boggling farmland belonged to the Aztecs as they adapted to their surroundings with Technology that we haven't yet matched with floating farmland. The Aztecian Government was a good system that perhaps doesn't quite parallel our own but was efficient because the Aztecs would simply conquer a territory then count it as it's own city-state which in exchange for taxes sent to the emperor would be protected by aztec army, if they refused of course it would be attacked by the aztec army, so the choice was simple for the city-states which still prospered under Aztec supervision. All was well until the arrival of a cabin boy on Colombus' ship. His name was Hernan Cortez the first of the conquistadors who held Montezuma ransom to his people for their precious gold and silver after he recieved the ransom he killed Montezuma just the same and before long their modern guns and diesease (especially smallpox) killed off the entire empire of millions in just a few years.
Perhaps the worst Government of the three empires was the Incas who had similar city-states but taxed them insanely, ironically when the city-states revolted the conquistadors came and the brand new leader Atuhwalpa who had conquered his twin in rivalry for leadership of the Incas, then came Francisco Pizzaro who was a cabin boy on Hernan Cortez's ship did what he saw Cortez do and held Atuhwalpa ransom for precious metals and killed him as well after ransom was met.
Of course all the empires had much more depth in all of these subjects and more but in the best interest of keeping this semi-overdone essay short.