  We use the terahertz (THz) emission spectroscopy to study femtosecond
photocurrent dynamics in the prototypical 2D semiconductor, transition metal
dichalcogenide MoSe$_2$. We identify several distinct mechanisms producing THz
radiation in response to an ultrashort ($30\,$fs) optical excitation in a
bilayer (BL) and a multilayer (ML) sample. In the ML, the THz radiation is
generated at a picosecond timescale by out-of-plane currents due to the drift
of photoexcited charge carriers in the surface electric field. The BL emission
is generated by an in-plane shift current. Finally, we observe oscillations at
about $23\,$THz in the emission from the BL sample. We attribute the
oscillations to quantum beats between two excitonic states with energetic
separation of $\sim100\,$meV.
