  We calculate the mass difference between the $\Upsilon$ and $\eta_b$ and the
$\Upsilon$ leptonic width from lattice QCD using the Highly Improved Staggered
Quark formalism for the $b$ quark and including $u$, $d$, $s$ and $c$ quarks in
the sea. We have results for lattices with lattice spacing as low as 0.03 fm
and multiple heavy quark masses, enabling us to map out the heavy quark mass
dependence and determine values at the $b$ quark mass. Our results are:
$M_{\Upsilon} -M_{\eta_b} = 57.5(2.3)(1.0) \,\mathrm{MeV}$ (where the second
uncertainty comes from neglect of quark-line disconnected correlation
functions) and decay constants, $f_{\eta_b}=724(12)$ MeV and $f_{\Upsilon}
=677.2(9.7)$ MeV, giving $\Gamma(\Upsilon \rightarrow e^+e^-) = 1.292(37)(3)
\,\mathrm{keV}$. The hyperfine splitting and leptonic width are both in good
agreement with experiment, and provide the most accurate lattice QCD results to
date for these quantities by some margin. At the same time results for the time
moments of the vector-vector correlation function can be compared to values for
the $b$ quark contribution to $\sigma(e^+e^- \rightarrow \mathrm{hadrons})$
determined from experiment. Moments 4--10 provide a 2\% test of QCD and yield a
$b$ quark contribution to the anomalous magnetic moment of the muon of
0.300(15)$\times 10^{-10}$. Our results, covering a range of heavy quark
masses, may also be useful to constrain QCD-like composite theories for beyond
the Standard Model physics.
