There are many threads on various internet forums asking about the details of these sliders, but no good answers. In this post, I endeavor to change that.
Structure is the usual "unsharp mask" sharpening used on every image editing program in existence. The twist is that the radius used scales with the size of the image. Large images mean bigger radii. This YouTube video contains proof:
https://www.youtube.com/watch?v=P8jNw-y_CeU.
Clarity is a bit trickier. Let's start with the "Neutral" option with positive values. Here, C1 performs a global adjustment of the luma curve. The shape of the curve is a shallow S which crosses the input = output line (45 degree line from lower left to upper left corner of the curves window) at an L value somewhere in the low 30s, where I measure in L in the LAB color space. (Note the luma curve window uses different units.) In particular, neutral does not touch any color information. The A and B values in LAB stay the same. You can prove this for yourself by going to DPReview, downloading one of their test images with a Kodak grey step wedge, and trying to match the results of a positive clarity adjustment with a luma curve.
With "natural" and "punch," there are adjustments to the A and B values to make the colors more saturated. "Punch" saturates more than "natural." I did not check, but my (totally ignorant) guess is that they're curving in the A and B spaces too, so you could plausibly recreate these sliders in another editing program (e.g. Photoshop) just by figuring out the right curves from a synthetic test image of many color patches.
A point that surprised me is that clarity (at least neutral with positive values) is a totally global adjustment. The Capture One documentation makes it sound like local contrast enhancement. It's not!
I don't know what negative Clarity values do, but it seems like a different algorithm. The "neutral" option still operates totally on the L channel, but there are now local effects that can be detected by measuring LAB values on the transition regions between color patches. I think. I can't actually see them, and there's noise in the LAB measurements, but it replicated enough with different patches that I'm pretty sure of it.
I hope this was helpful, or at least interesting.