Fix a hash function $H\colon \{0,1\}^m \to \{0,1\}^n$ built out of a sponge of capacity $c$.
To raise the cost of both generic collision and generic preimage searches on classical or quantum computers above $2^\lambda$, pick $c = n = 2\lambda$: the capacity and hash length need only be double the security target, no more.
For example, for a 128-bit security level even against a quantum adversary, it suffices to pick $c = n = 256$. Out of an abundance of caution, and partly for political reasons arising from the timing so soon after Snowden and Project BULLRUN[1], NIST chose $c = 2n = 4\lambda$ instead, e.g. $c = 512$ for SHA3-256 giving a 128-bit security level, at substantial—and at the time politically justifiable but now technically regrettable—cost to performance. In contrast, the newer SHAKE128 and SHAKE256 were unhampered by political concerns by virtue of being novel anyway, and follow the $c = 2\lambda$ formula.
If this advice seems at odds with the conventional wisdom that you have to double or triple hash sizes to maintain the same security against quantum adversaries as you had against classical adversaries, let's review the state of the art in generic attacks.
Collision resistance
The conventional wisdom outside cryptographers is that the best you can do classically is create a large table of outputs for random inputs and hope you find a collision eventually by the birthday paradox, which would cost $O(2^{n/2})$ time and $O(2^{n/2})$ space to store the table. But you don't need to store a table—a variant of Pollard's $\rho$[2] can do it in constant space.
So the best generic classical collision search algorithm is van Oorschot–Wiener[3] with cost $O(2^{n/2})$, which can be parallelized to give a linear speedup—e.g., parallelize it $O(2^{n/4})$ ways and get an answer in $O(2^{n/4})$ time at the same cost of $O(2^{n/2})$. For a sponge, the same kind of generic birthday attack works on the $c$-bit hidden state at cost $O(2^{c/2})$.
The best generic quantum collision algorithms are advertised to take $O(2^{n/3})$ time, but this ignores the reality that there are costs other than time[4]. The Brassard–Høyer–Tapp algorithm[5] works by building a table of size $O(2^{n/3})$ and then running Grover's algorithm for $O(2^{n/3})$ steps, with an area*time cost of $O(2^{2n/3})$, far costlier than a classical search can be. The Ambainis algorithm[6] can be adapted into a collision search algorithm for similar cost $O(2^{2m/3})$. There are some other variations on the theme but none of them have broken the $O(2^{n/2})$ cost barrier even under the wildly unrealistic premise that qubit operations are as cheap as bit operations and communication is free[7]. For a sponge, the same kind of generic birthday attack works on the $c$-bit hidden state at cost $O(2^{2c/3})$[8].
There is a lower bound $\Omega(2^{n/3})$ on the number of queries a generic quantum algorithm must make to $H$[9], so if there is any advance in this area, it must be dramatic reductions in storage costs.
The best generic (sponge) collision attacks are classical algorithms which can be run today—there is no additional concern from quantum algorithms.
Preimage resistance
The conventional wisdom outside cryptographers is that the best you can do classically is to try an expected $O(2^n)$ inputs before you'll find one that matches. But you can save cost across multiple targets with Oechslin's rainbow tables[10] on a parallel machine[11], attaining a cost of $O(2^n/t)$ in the time for $O(2^n/t^3)$ sequential evaluations of $H$, if parallelized $t^2$ ways to find a preimage for the first of $t$ targets. For a sponge, the same generic preimage attacks apply to recovering the $c$-bit hidden state at cost $O(2^c/t)$ in time $O(2^c/t^3)$.
The best generic quantum preimage algorithm, Grover's algorithm[12], is advertised to take $O(2^{n/2})$ time, and it doesn't take much space, so the area*time cost is a somewhat scary-looking $O(2^{n/2})$—but even if you could evaluate $H$ once per nanosecond sequentially without time for any other computation, it would take half a millennium to get an answer. And Grover, it turns out, doesn't parallelize well or give much of a batch advantage[13]: spending $p$ times as much energy to run Grover on $p$ machines in parallel gives only a factor of $\sqrt{p}$ speedup, so the net cost grows by a factor of $\sqrt{p}$, and a batch attack on $t \ll p$ targets gives only a factor of $t^{1/4}$ speedup. For a sponge, the same generic preimage attacks apply to recovering the $c$-bit hidden state in time $O(2^{c/2})$ with similar costs for parallelism and modest batch advantage.
There is some additional concern for speedups in preimage attacks attainable by quantum computers, but you should also be worried about multi-target batch attacks using classical computers too.
Second-preimage resistance
There's no generic second-preimage attack that works better than a generic preimage attack, and collision resistance implies second-preimage resistance anyway, so there's not much more to say about this.
