  Finding physical principles lying behind quantum mechanics is essential to
understand various quantum features, e.g., the quantum correlations, in a
theory-independent manner. Here we propose such a principle, namely, no
disturbance without uncertainty, stating that the disturbance caused by a
measurement to a subsequent incompatible measurement is no larger than the
uncertainty of the first measurement, equipped with suitable theory-independent
measures for disturbance and uncertainty. When applied to local systems in a
multipartite scenario, our principle imposes such a strong constraint on
non-signaling correlations that quantum correlations can be recovered in many
cases: i. it accounts for the Tsirelsons bound; ii. it provides the so far
tightest boundary for a family of the noisy super-nonlocal box with 3
parameters, and iii. it rules out an almost quantum correlation from quantum
correlations by which all the previous principles fail, as well as the
celebrated quantum criterion due to Navascues, Pironio, and Acin. Our results
pave the way to understand nonlocality exhibited in quantum correlations from
local principles.
