That player loses half their life after combat damage has been subtracted from the player's life total. The amount of life the player loses is determined as the triggered ability resolves.
If multiple Quietus Spikes trigger at the same time, that player loses half their life when the first ability resolves, then loses half of the remainder when the next ability resolves, and so on. The player does not lose the same amount each time.
In a Two-Headed Giant game, after combat damage is dealt, Quietus Spike looks at that player's life total (which is the same as the team's life total) when determining how much life the team will lose, which basically means the team's life total is halved. Here's an example of the math: The team has 19 life, so the player has 19 life. Quietus Spike causes the team to lose 10 life (19 divided by 2, rounded up). The team's life total becomes 9 (19 minus 10).