Luminarch Ascension's first ability doesn't trigger if you have lost life before an opponent's end step begins. If it does trigger but you lose life before it resolves, it does nothing and you won't put a quest counter on Luminarch Ascension.

Luminarch Ascension's first ability cares only whether you lost life this turn, even if Luminarch Ascension wasn't on the battlefield when that happened. It doesn't care how much you lost, whether you also gained life, or even whether you gained more life than you lost.

You can activate Luminarch Ascension's last ability during the end step in which it receives its fourth quest counter.