On my opponent's turn, during main phase 2, I Quicken a Seize the Day. My opponent gets an additional combat phase, correct? One of my friends is convinced that the caster of the Seize the Day "controls" the extra combat, and so can attack on an opponents turn, but I'm pretty sure this is not the case. Clarification?
One of the rulings on Seize the Day addresses this situation:
If you somehow cast Seize the Day during an opponent’s main phase, that opponent attacks during the resulting combat phase, since it’s still that player’s turn. If you somehow cast it not during a main phase at all, all that happens is you untap the target creature.
The wording of Seize the Day is pretty straightforward:
Untap target creature. After this main phase, there is an additional combat phase followed by an additional main phase.
It just says "there is another combat phase", not "you get another combat phase". There is no concept of "owning" a phase, other than whose turn it's on.
506.2. During the combat phase, the active player is the attacking player; creatures that player controls may attack. During the combat phase of a two-player game, the nonactive player is the defending player; that player and planeswalkers they control may be attacked.
If you cast it on your opponent's turn, then your opponent is the active player, and so they are the attacking player.