  We study femtosecond spin currents through MgO tunneling barriers in CoFeB(2
nm)|MgO($d$)|Pt(2 nm) stacks by terahertz emission spectroscopy. To obtain
transport information independent of extrinsic experimental factors, we
determine the complex-valued spin conductance $\tilde{G}_d (\omega)$ of the MgO
layer (thickness d= 0-6 {\AA} over a wide frequency range $(\omega/2\pi=$ 0.5-8
THz). In the time $(t)$ domain,$ G_d (t)$ has an instantaneous and delayed
component that point to (i) spin transport through Pt pinholes in MgO, (ii)
coherent spin tunneling and (iii) incoherent resonant spin tunneling mediated
by defect states in MgO. A remarkable signature of (iii) is its relaxation time
that grows monotonically with $d$ to as much as 270 fs at $d= 6$ {\AA}, in full
agreement with an analytical model. Our results indicate that terahertz spin
conductance spectroscopy will yield new and relevant insights into ultrafast
spin transport for a wide range of materials.
