Jeremy, go ahead, throw those potatoes into the pile, but restrict it to peelings and skins.
Potato peels and skins rot down quickly without any risk of regrowth.
Its the whole potatoes that may re-grow if you throw them into the mix.
Better to mash them or cut them up in little pieces.
They would rot down much faster in any case.
You certainly wouldn't want potatoes growing in that compost pile -- at least, not without informing guests that the white stuff with gravy comes from it.

I'll cut them up from now on,I started to notice whole potatoes still in the pile when I mixed it, sometimes weeks later. I was really wondering about putting the potatoes directly onto the future pumpkin patch, under leaves, just to save the time from moving the compost to the patch (not really that much hastle, just seeing if my idea was any good).

Whole potatoes are no more likely to sprout than the skins because new growth comes from the eyes that are right on the surface, the skin, of the potatoes. Seed potatoes are a an eye with a bit of the pulp attached, but I have had those eyes sprout even with none of the pulp attached. As long as you are willing to, maybe, work with germinating potatoes there is no reason why you could not add those potatoes. If they are added now and not buried too deeply the cold weather may well destroy any desire on the part of the potatoes to sprout.

You'd have more than just a few sprouts to pull up.

Laying out potato tubers, whether whole or chunked, and then covering them with leaves - that's how I plant my potato patch. You'd have some losses, from rot and rodents, but you'd basically have a potato patch next spring. Plus all the fungal, bacterial and viral diseases that can spread on potato tubers.

I'd chop and compost them myself. Cut up small enough, and heated up enough most of the tuber would decompose rather than sprout, and if the pile got hot enough you'd greatly reduce the fungal and bacterial diseases at least.

I know it's more work, but do you want pumpkins and gourds, or do you want potatoes?

Has anyone ever cut the potatoes up into small pieces then boiled them? It seems to work for me after cooking them.

There is nothing better than early new potatoes. Last spring I had some volunteer potatoes that provided little nugget new potatoes from May on...boiled with mint they are delicious! Why don't you try an area with potatoes for eating?

Whe I get a bunch of taters to compost, I microwave them b4 throwing them in the compost pile. Never had one sprout so far.