Why exactly does **quadratic probing** lead to a shorter avg. search time than linear probing?

I fully get that **linear probing** leads to a higher concentration of used slots in the hash table (i.e. higher “**clustering**” of used consecutive indices). However, it’s not immediately trivial (to me at least) **why** that translates to **higher search times** in expectation than in quadratic probing, since in both linear and quadratic probing the first value of the probing sequence determines the rest of the sequence.

I suppose this has to do more with the probability of collisions **between** different probing sequences. Perhaps different auxiliary hash values are less likely to lead to collisions early in the probing sequence in quadratic than in linear hashing, but I haven’t seen this result derived or formalized.