At the quantum level, time has no directionality. All of the processes act as though they obey no preferred ordering in time. Some virtual processes even allow particles to move in zigzag paths backwards and forwards in time.
Processes evolve in the direction of increasing their level of disorder or 'entropy'.
At this time, no one knows where the Thermodynamic Arrow of Time emerges from a time-symmetric quantum level, or even at the molecular scale for that matter. There is also, at the largest scale, the Cosmological Arrow of Time which is determined by the fact that the universe is expanding. Some cosmologists such as Stephen Hawking believe that the Thermodynamic Arrow of Time and the Cosmological Arrow of Time are linked together, but this leads to the curious results that if the universe is destined to collapse in the future, the Cosmological Arrow of Time will be reversed, entropy will begin to decrease, and systems will evolve to ordered states rather than disorderd ones. We don't really know, and I do not fully understand a lot of the arguments. We don't know why this arrow points in the opposite direction of the Thermodynamic Arrow except that in a closed system like the universe, order will emerge in some collections of matter in the universe at the expense of increasing the entropy in other parts of the 'system'.