It exists
The Quantum Barbarian hinges on the Berserker option (which causes him to start raging below a certain amount of hit points, and stop above that amount) and the Bear Warrior prestige class. Bear warriors turn into bears when they rage, gaining a certain amount of hit points in the process that they lose when the rage ends. As a result, if the berserker dips into their rage threshold within the amount given by Bear Warrior, they get caught in a cycle of rage-start-rage-end that won't end until or unless their HP drops below or above that threshold.
Beserker Strength (PHB2): When HP falls below Barbarian level x5, you enter rage. There is no limit to how many times this can happen. This rage can only end when you fall unconscious, helpless, or receive healing to bring your current HP above the threshold.
Bear Warrior (Complete Warrior Prestige Class): When you rage, you can choose to enter a bear form. This provides some extra bonuses, the important one being a +4/+8 (depending on level) CON bonus.
So:
- You fall below the HP threshold, and rage activates.
- As a Bear Warrior, rage increases your CON.
- This increase in CON raises your HP above the limit, so rage ends.
- Rage ending removes the extra CON, lowering your HP below the threshold.
- Go To 1.
As mentioned, this wouldn't work by RAW due to the part about rage ending when you get healing to raise you above the limit. A CON increase is not technically healing, so #3 in the above sequence wouldn't happen by RAW. It's also worth noting that Berserker Strength is technically not "Rage" and thus may not qualify for Bear Warrior at all (though your DM may not get fussy over that detail).
No the Barbarian cannot Rage through a Trance, regardless of whether Trance would end your Rage or Not
Persistent Rage:
Beginning at 15th level, your rage is so fierce that it ends early only if you fall unconscious or if you choose to end it.
Note that Rage still has a defined end condition.
Rage:
Your rage lasts for 1 minute. It ends early if you are knocked unconscious or if your turn ends and you haven't attacked a hostile creature since your last turn or taken damage since then. You can also end your rage on your turn as a bonus action.
Since it seems unreasonable to continue using bonus actions every minute to keep raging while in a Trance, it wouldn't be possible to continue raging and trance at the same time.
Best Answer
I don't see why not...
So the PHB says:
So from a RAW perspective I'd probably say no because to enter a rage you most likely cannot already be in a rage. That would be similar to trying to enter a house that you are already inside of if interpreted literally. And I think that the fact that you can forcefully end your rage (essentially exiting it) supports this.
However...
From a RAI perspective I think that it is okay to extend your rage duration by expending another rage slot. Consider this:
I believe the emphasis for this lies in that rest is required to replenish a barbarians fury. However, until depleted, that reservoir can continue to be tapped. One does not simply stop being furious because a minute has passed. The main thing to take into account with this mechanic is the limited reservoir of fury.
Consider also, the rage timer could reach its end, kicking the barbarian out of rage at the beginning of their turn, only for them to immediately pop another rage. There is technically no significant amount of time between the first and second rage. It would not make sense for the barbarian to experience a single moment of clarity spanning the bridge of one solitary second before becoming consumed by a furious rage once more.
In summary:
RAW is most likely a no, but from the perspective of RAI I'd give it a yes.