I haven't made the effort to collect stock exchange and Bitcoin price data yet. It's just my personal (subjective) observation. I might do it when I have more time at my disposal.
I don't know if you're already looked through the internet or if you really want to do this by yourself, but data about this already exist:
https://www.investopedia.com/news/are-bitcoin-price-and-equity-markets-returns-correlated/There are a lot more, and the consensus seems to be that there is
some correlation. As for your original question, I would think that the entry of institutional investors would only serve to increase correlation. It really hasn't happened en masse yet, so you could probably still use Bitcoin as a hedge against the stock market right now, but probably not for much longer. It's pretty sad if you think about it.