Right now there is no correlation between the two, but it can change in the future if Bitcoin attracts the institutional investors and the HNWIs. Take the case of gold. Gold prices go up, when that of the equities go down. That is because gold is being viewed as a safe asset with low volatility (at least when compared to the equities).
Yet gold practically in tandem with equities, up until 2010/11 and QE to infinity, from which point the equities have since soared, and gold has crashed to miserable depths.
Financial establishment policy for the time being, is keep equities high, gold and other commodities low and I wouldn't like to take a bet against them. Betting against them is betting that they fuck up, which I suppose could happen. Gold and commodites either go up when financial establishment change their policy, or when/if they lose control.