When can I quick save?

My key bindings tell me that F5 is "Quick Save". However, I press it after defeating a tough enemy in combat and nothing happens. No UI element tells me it saved.

I experimented and tried a F9 quick load, and it took me back to my last auto save. When can I quick save?

You can quicksave when there aren't hostiles in your immediate vicinity.

The game blocks you from saving during some sequences, but you should be able to quicksave freely the rest of the time.

Sadly, there's no real indicator of when it will work. Your best bet, is to do what I do, and if you think you might be able to quicksave, hit the button at every opportunity. It'll work enough times to minimize your backtracking.

You can only quicksave when you are not in combat and there are no enemies around. If you are in combat, the game will not let you quicksave, unfortunately, and if there are still enemies left, you will have to wait. Some areas of the game will also block this as well.

You may have to wait a certain amount of time until after there are no more enemies in order to do so. In earlier games, this was when your squaddies yelled "All clear!" but I haven't noticed the same functionality in this game.