  The Large Synoptic Survey Telescope (LSST) will open a discovery frontier for
faint and fast transients with its ability to detect variable flux components
down to $\sim$24.5 mag in a $\sim$30 second exposure. Spectroscopic follow-up
of such phenomena - which are necessary for understanding the physics of
stellar explosions - can require a rapid response and several hours with a
8-10m telescope, making it both expensive and difficult to acquire. The future
Extremely Large Telescopes (ELTs) would be able to provide not only
spectroscopy but capabilities such as spectropolarimetry and high-resolution
diffraction-limited imaging that would contribute to future advances in our
physical understanding of stellar explosions. In this white paper we focus on
several specific scientific impacts in the field of explosive transient
astrophysics that will be generated by the combination of LSST's discovery
abilities and ELTs' follow-up capacities. First, we map the uncharted frontier
of discovery phase-space in terms of intrinsic luminosity and timescales for
explosive transients, where we expect the unexpected. We then focus on six
areas with open science questions for known transients: the progenitors of
thermonuclear supernovae (SNe), mass loss prior to core collapse, asymmetry in
stellar explosions, light echoes, high-$z$ transients, and strongly lensed SNe.
We conclude with a brief discussion of the practical aspects of ELT & LSST
synergy.
