Suisei no Gargantia, Terra Formars Recommendations
Both anime take place in a future where humans are fighting highly evolved alien enemies.
The fight is to determine the more dominant species and the loser is just vermin.
Suisei's future is far after humans had abandoned earth (thousands and colonized far out into the universe. Their ships shoot lasers, warp through space, and have sentient AI.
However, the main story of this takes place in a lost colony and is relatively lighthearted as our space fleet main learns to communicate with a lost culture, until he finds and has to fight the alien species again... and learn their true origin.
Terra Formar's future is 500 years