Does hydrochloric acid conduct electricity in its molten or solid state?
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First of all we should distinguish between hydrochloric acid and hydrogen chloride. Pure HCl is called hydrogen chloride and it is a gas at standard temperature and pressure. When the gas is dissolved in water, we call that solution hydrochloric acid. So hydrochloric acid is an aqueous solution of HCl in water.
The conductance of electricity by chemicals is based on the presence of ions. Ions are highly charged species that can readily conduct an electrical current. So hydrochloric acid, since it exists as ions in water, readily conducts electricity. But hydrogen chloride the gas does not exist as ions since there is no solvent medium for them to dissolve into. So hydrogen chloride gas does not conduct electricity. If you were to freeze the gas into a solid, it still would not conduct electricity for the same reason. I'm not sure if molten HCl can actually exist but it too would not conduct electricity.
Pure HCl is a gas boiling at -85 degree Celsius. It is a predominantly covalent compound with ~ 17% ionic character. This implies that there should be very few ions present in it in liquid state and therefore, it should not conduct electricity effectively in liquid state, let alone the question of conducting electricity in solid state. What we see in laboratories is an aqueous solution of the pure gaseous HCl, of varying concentrations. In such solutions, high hydration energy renders the necessary bond dissociation energy for splitting the H-Cl bond and thus it exists in fully ionized state, that’s why it is termed a strong acid. Owing to the presence of ions, it can conduct electricity very well in aqueous solution.
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