  The aim of this paper is to provide complementary quantitative extensions of
two results of H.S. Shapiro on the time-frequency concentration of orthonormal
sequences in $L^2 (\R)$. More precisely, Shapiro proved that if the elements of
an orthonormal sequence and their Fourier transforms are all pointwise bounded
by a fixed function in $L^2(\R)$ then the sequence is finite. In a related
result, Shapiro also proved that if the elements of an orthonormal sequence and
their Fourier transforms have uniformly bounded means and dispersions then the
sequence is finite. This paper gives quantitative bounds on the size of the
finite orthonormal sequences in Shapiro's uncertainty principles. The bounds
are obtained by using prolate sphero\"{i}dal wave functions and combinatorial
estimates on the number of elements in a spherical code. Extensions for Riesz
bases and different measures of time-frequency concentration are also given.
