  We propose an uncertainty-complementarity balance relation and build
quantitative connections among non-locality, complementarity, and uncertainty.
Our balance relation, which is formulated in a theory-independent manner,
states that for two measurements performed sequentially, the complementarity
demonstrated in the first measurement in terms of disturbance is no greater
than the uncertainty of the first measurement. Quantum theory respects our
balance relation, from which the Tsirelson bound can be derived, up to an
inessential assumption. In the simplest Bell scenario, we show that the bound
of Clauser-Horne-Shimony-Holt inequality for a general non-local theory can be
expressed as a function of the balance strength, a constance for the given
theory. As an application, we derive the balance strength as well as the
nonlocal bound of Popescu-Rohrlich box. Our results shed light on quantitative
connections among three fundamental concepts, i.e., uncertainty,
complementarity and non-locality.
