Do not fold a mattress, ever. This will damage the interior structure and ruin the mattress.
Do not cut the box spring in half to fit it up the staircase.
A spiral staircase can present a difficult problem for movers, and ideally you should measure your larger furniture before moving to make sure it will fit. If you have a bed that needs to get up to the top of a spiral staircase, learn a few maneuvers to make the process easier.
Start with the box spring because this will most likely be the most difficult. Lay it over the open end of the staircase. Have a helper assist you in moving it upward. When you get to the top, you may need to stand it upright to get onto the landing.
Stand the box spring upright if it does not fit over the edge of the rail and have your helper assist you in carrying it up. This could present a challenge in cramped spaces.
If your queen bed is still too difficult to get up the spiral staircase, consider switching to a king, which comes with two box springs pushed together.
Lay your mattress horizontally, and carry it up that way. You should be able to curve it gently to get it to fit around the center of the stairs.
Look for a balcony. If one is available, it is possible that that the bed can be raised up with ropes, pulleys and lots of help.