  The quest for fundamental test of quantum mechanics is an ongoing effort. We
here address the question of what are the lowest possible moments needed to
prove quantum nonlocality and noncontextuality without any further assumption
-- in particular without the often assumed dichotomy. We first show that second
order correlations can always be explained by a classical noncontextual
local-hidden-variable theory. Similar third-order correlations also cannot
violate classical inequalities in general, except for a special state-dependent
noncontextuality. However, we show that fourth-order correlations can violate
locality and state-independent noncontextuality. Finally we obtain a
fourth-order continuous-variable Bell inequality for position and momentum,
which can be violated and might be useful in Bell tests closing all loopholes
simultaneously.
