We give quantum algorithms for exponential-time problems on $n$ vertices or elements, with bounded error, polynomial space and no quantum random access memory. Our results are these. (1) 4-Colouring in quantum time $O(1.3032^n)$, from a polynomial-space branching algorithm with polynomial-space terminal algorithms; with the branching algorithm alone, $O(1.3100^n)$. The previous best bound in this model, to our knowledge, is $O(1.3231^n)$. (2) $k$-Colouring below the previous best bound for $k \in \{5, 7, 8, 9, 10, 11\}$ and $13 \le k \le 18$, from $O(1.4518^n)$ for $k = 5$ to $O(1.9233^n)$ for $k = 18$. (3) Dominating Set in $O(1.2235^n)$, against the previous $O(2^{n/2})$, Independent Dominating Set in $O(1.1583^n)$, and Edge Dominating Set and Minimum Maximal Matching in $O(1.1501^n)$. (4) A quantum form of monotone local search: if the extension problem has a backtracking tree with $c^k$ nodes up to polynomial factors, the problem is solved in quantum time $(2 - 1/c)^{n/2}$ up to polynomial factors. This gives Feedback Vertex Set in $O(1.3081^n)$, Subset and Group Feedback Vertex Set in $O(1.3229^n)$, Cluster Vertex Deletion in $O(1.2033^n)$, 3-Hitting Set in $O(1.2405^n)$, Proper Interval Vertex Deletion in $O(1.3541^n)$, Multicut on Trees in $O(1.1647^n)$ and Min-Ones $d$-SAT in $O^*((2-1/d)^{n/2})$. (5) Quantum exponential-time approximations below the quantum exact bounds, such as a 1.1-approximation for Vertex Cover in $O(1.0555^n)$ and for Feedback Vertex Set in $O(1.2161^n)$. The colouring exponents for $k \ge 6$ come from an evaluation of a known reduction recursion, computed in floating point with a margin.