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.