I have a 50 AMP breaker at main panel. I am currently installing a local 50 AMP GFCI control box 5 feet from the spa. The 80 foot run of existing wiring from the main panel directly to the spa is 2 hot black #6 AWG, 1 white #10 AWG and 1 green ground #10 AWG in 3/4 inch gray PVC conduit. Are these wiring sizes correct or should they all be the same gauge? Can the wiring from the local box to the spa control box be the same size as what is coming from the main?

Your hots and neutral need to be the same size, the ground can be one size smaller. I.e. if you use 6 ga hots, use 6 ga neutral, and 8 ga ground. Your conduit is too small for such wires as well.