The correlation I heard most about in economics/finance was that stock prices and bond yields were negatively correlated; as the stock market does better, bond yields fall (company's doing well as evidenced by stocks, so it's a good credit risk, so YTM of its bonds on the market goes down).
The correlation, if any, between the stock and futures market should be visible in the actual price histories. Index prices may be useful, but what's more likely is that various future prices have correlation with various companies' stocks. Where the future reflects the price of a raw material that is a significant cost of goods sold for a company, you'll see these two move inversely to each other in the short term.
I think that if there is a causative relationship here, its that futures prices influence stock prices, not the other way around. The futures market generally represents the cost side of a consumer goods producer's bottom line. The stock market represents its profits. As futures go up, profit expectations go down, putting pressure on stock prices. Industries that deal in services, or in other types of goods, can still be affected because a rise in the cost of something consumers need will cause them to spend less on other things which affects margins in those other areas. So, in the short and medium term, when the futures market goes up the stock market sees a dip, and vice versa.
However, companies adapt; they can put upward pressure on prices for their goods to restore their desired margins, usually by slowly increasing them to prevent sticker shock (though elasticity of demand plays a part; the more we need something no matter what it costs, the faster prices can increase). To maintain costs, they can make things cheaper using less expensive materials (more plastic, less steel). They can restructure production processes (translated: move factories offshore, or at least to "right-to-work" states with less union strength) to save costs elsewhere. All of these reduce costs and thus increase profits, but take time to implement. Many of these things reduce direct costs, reducing demand for the commodity and causing the futures prices to go back down. So, over the long term, these differences even out, and it's down to the things that affect the entire market (inflation, consumer/investor confidence, monetary policy).