Generalised 3SUM (k-SUM) problem?
$k$-SUM can be solved more quickly as follows.


For even $k$: Compute a sorted list $S$ of all sums of $k/2$ input elements.  Check whether $S$ contains both some number $x$ and its negation $-x$.  The algorithm runs in $O(n^{k/2}\log n)$ time.
For odd $k$: Compute the sorted list $S$ of all sums of $(k-1)/2$ input elements.  For each input element $a$, check whether $S$ contains both $x$ and $-a-x$, for some number $x$.  (The second step is essentially the $O(n^2)$-time algorithm for 3SUM.)  The algorithm runs in $O(n^{(k+1)/2})$ time.


Both algorithms are optimal (except possibly for the log factor when $k$ is even and bigger than $2$) for any constant $k$ in a certain weak but natural restriction of the linear decision tree model of computation.  For more details, see:


Nir Ailon and Bernard Chazelle.  Lower bounds for linear degeneracy testing.  JACM 2005.
Jeff Erickson.  Lower bounds for linear satisfiability problems.  CJTCS 1999.