Shed Sub Panel Wire Thickness 150 ft
#4 THWN aluminum is exactly what I'd use.
As far as voltage drop, 150' isn't far enough for voltage drop to become a concern.  I don't even run the numbers until 180' :)
#4 THWN aluminum breakers at 70A (it's rated for 65A but you round up to the next breaker size which exists).
Because it's capable of 70A, you need a 70A ground wire, which is #8 copper or #6 aluminum.   If aluminum, it must be insulated.  You're allowed to use #4 aluminum, and the benefit is that you get to use black wire and mark it green with phase tape.  #8 or #6 ground would need to be actually green or bare.