The ITP A1 EOS is roughly equivalent to the Quark Mini 123 except the latter's XP-G is brighter, esp. on 16340 - becomes a "wall of light". Similarly for ITP A2 EOS and Quark Mini AA. For something even smaller, the ITP A3 EOS Upgrade Edition runs on AAA or 10440 (this one has no Quark equivalent).
These lights work fine on lithium-ion cells like 14500 or 16340 or 10440, they will handle the voltage, the problem is that they are so small that they simply can't dissipate the heat if run on high for more than a short while. So as long as you just use short bursts on high (less than a half minute or minute), or heat sink them somehow, all is fine.
Another nice small light (AAA/10440) would be the LiteFlux LF2XT (clicky) or LF2X (twisty - older model, look in marketplace), they have an amazing feature set.
Since your first post mentioned throw, you might look at the small throwers such as the Lumapower D-Mini (several models) or JetBeam RRT-0 or Jet-II Pro IBS.