how long does a miner have to hold bitcoin after winning it…

A miner can sell their Bitcoin as soon as they receive it. There's no mandatory holding period. However, the miner has to wait for the mined block to get confirmed by other nodes in the network, which usually takes about 10 minutes. After that, they can freely sell or spend the Bitcoin.

isnt there a wait period from the moment a miner guesses the nonce to the moment they gain control of the mining reward

Yes, you're correct. When a miner successfully mines a new block by guessing the nonce, they have to wait for 100 more blocks to be added to the blockchain before they can spend the mining reward. This is approximately 16 to 17 hours, considering the average block time for Bitcoin is about 10 minutes. This is known as coinbase maturity in the bitcoin world and is in place to ensure the stability and security of the blockchain.

how does it ensure stability?

The 100-block maturation period for newly mined Bitcoin provides a buffer against the possibility of a blockchain reorganization.

Sometimes, two miners may solve a block at nearly the same time, and until the network resolves which block to continue building upon, there's a split with two potential versions of the blockchain. This is known as a blockchain fork.

If a miner were to spend their rewards immediately, and then a reorganization occurred where the miner's block was no longer part of the longest chain (and thus, their reward invalidated), it could create a double-spending problem.

The longer the blockchain confirmation time (i.e., the more blocks added after the miner's block), the less likely a reorganization will affect the miner's block. After 100 blocks, it's almost certain that the block is permanently part of the blockchain. This mechanism ensures the security and stability of the system.