In which state of matter does heat travel fastest?

Matter in any state can be propelled as fast as you want.

Answer:

Within the relativistic Universe matter can go as fast as the speed of light (with great difficulty). Excluding tachyons which (if they exist) the only matter observed to go at the speed of light are photons. Photons do not fall into the normal solid, liquid or gas states of matter as they have both wave and particle characteristics and no rest mass.

For normal matter, the problems with acceleration require that the lightest particles will travel fastest for any amount of acceleration energy. This would call for the dissolution of the matter into individual atoms (into a gas) and the acceleration of the individual atoms.

In this roundabout manner gases can be accelerated the most so the can travel the fastest.