  It is shown that nearly-flat 3+1D spacetime emerging from a dual quantum
field theory in 2+1D displays quantum fluctuations from classical Euclidean
geometry on macroscopic scales. A covariant holographic mapping is assumed,
where plane wave states with wavevector k on a 2D surface map onto classical
null trajectories in the emergent third dimension at an angle \theta=l_P k
relative to the surface element normal, where l_P denotes the Planck length.
Null trajectories in the 3+1D world then display quantum uncertainty of angular
orientation, with standard deviation \Delta\theta=\sqrt{l_P/z} for longitudinal
propagation distance z in a given frame. The quantum complementarity of
transverse position at macroscopically separated events along null trajectories
corresponds to a geometry that is not completely classical, but displays
observable holographic quantum noise. A statistical estimator of the
fluctuations from Euclidean behavior is given for a simple thought experiment
based on measured sides of triangles. The effect can be viewed as sampling
noise due to the limited degrees of freedom of such a theory, consistent with
covariant bounds on entropy.
