[DrPete]

The alternating possession throw-in never happened for the "jump ball" the official called as time was expiring, so the arrow should not have been changed. You start the next quarter with the AP throw in according to the arrow.
And, just because Team A got the ball for the throw in to start the third quarter, doesn't mean Team B will get it for the fourth quarter, there are lots of ways the alternating arrow can change over the course of the game, not just at the beginning of a quarter.