Step 1: What is the Lucas test.
The Lucas test tells apart primary, secondary and tertiary alcohols. It uses concentrated HCl with zinc chloride, called the Lucas reagent.
Step 2: How the test shows results.
A tertiary alcohol reacts at once and turns cloudy straight away. A secondary alcohol reacts in a few minutes. A primary alcohol shows no change at room temperature.
Step 3: Why tertiary is fastest.
Tertiary alcohols react fast because they form a stable carbocation easily. This makes the cloudy alkyl chloride appear right away.
Step 4: Check the choices.
2-methylpropan-2-ol has three carbon groups on the carbon with $\text{OH}$, so it is a tertiary alcohol.
Step 5: Check the rest.
Propan-1-ol and ethanol are primary alcohols, so they show no quick change. Propan-2-ol is secondary, so it reacts only slowly.
Step 6: Final choice.
Only the tertiary alcohol reacts at once, so the answer is 2-methylpropan-2-ol.\[ \boxed{\text{2-Methylpropan-2-ol}} \]