ABSTRACT: Local realism implies constraints on the statistics of two physically separated systems. These constraints, known collectively as Bell inequalities, can be violated by quantum mechanics. The standard Bell inequalities apply to a pair of two-state systems and constrain the value of some linear combination of correlation functions between the two systems. The authors generalize these standard Bell inequalities in two ways. First, they 'chain' the Clauser-Horne-Shimony-Holt Bell inequality to obtain chained correlation Bell inequalities for two-state systems; they model a real experiment to show that these chained Bell inequalities lead to stronger quantum violations. Second, they formulate information-theoretic Bell inequalities, which are written in terms of the average information obtained in several measurements on a pair of physically separated systems.