To unlock the 4th round (2nd round of World Tournament), beat Max, Michael and Ray. You'll be able to fight Kai, Daichi, Tala and Robert. Beat them and you get to battle a dude from the BBA.
Defeat the first 4 characters 3 time to unlock the next round. To unlock the third round defeat the white tigers (the 4 characters from the second round) to unlock the third round, with Max and Ray from the Bey-Breakers and Michael from the All Stars.