Wire size. 125 feet underground to 100 amp sub panel
At 125’ 4 awg copper would provide 1.79% voltage drop using 4 gauge wire would allow for 70 amp breaker in your main panel. Since this is detached you will want a ground rod also. To save on wire cost you could use #2 aluminum and still have 70a breaker at the main.

If you would like to play with wire sizes google southwire voltage drop calculator, that’s what I use the only part that’s not obvious is the box that asks for parallel sets put 1 it will give you the wire size and voltage drop info, it’s a handy tool.