Trade off between time and query complexity
Here's an attempt at creating an artificial function with the following property:


The problem can be solved with $O(\log n)$ queries, but requires $\exp(n)$ time.
The problem can be solved with $O(n)$ time but that requires $O(n)$ queries


Let the input size be $n + \log n$. Let the first $\log n$ bits (let's call this string x) encode the input to a problem complete for EEXP. The next $n$ bits (let's call this string y) have the property that they are all zero if and only if x is a NO instance of the EEXP-complete problem.

In words, the first $\log n$ bits encode a hard problem, and the next $n$ bits give you a clue about the solution of the problem. However, to figure out the solution by looking at the $n$ bit string you make $\Omega(n)$ queries.

So this problem can be solved by either reading only the first $\log n$ bits and spending exp(n) time or by reading $n$ bits and using only linear time.

The same function goes through for quantum query complexity.. insert square root signs where necessary.