This paper presents efficient constructions of unitary $k$-designs and parallel-secure pseudorandom unitaries (PRUs) using quantum algorithms that lift random permutations over $\boldsymbol{S}(N)$ to random unitaries over $\boldsymbol{U}(N)$ for $N = 2^n$. The main results are achieved by showing that products of exponentiated sums of random permutations with random phases approximately match the first $2^{\Omega(n)}$ moments of the Haar measure. By substituting either $\tilde{O}(k)$-wise independent permutations or quantum-secure pseudorandom permutations (PRPs) in place of the random permutations, the authors obtain efficient constructions of unitary $k$-designs and parallel-secure PRUs. The key technical step involves demonstrating that an orthonormal basis for irreducible representations of the partition algebra has a low-degree large-$N$ expansion, allowing the distinguishing probability to be expressed as a low-degree rational polynomial of the dimension $N$. The paper also discusses the connection between the large-$N$ expansion in random matrix theory and the polynomial method, which is crucial for proving query lower bounds at finite-$N$.This paper presents efficient constructions of unitary $k$-designs and parallel-secure pseudorandom unitaries (PRUs) using quantum algorithms that lift random permutations over $\boldsymbol{S}(N)$ to random unitaries over $\boldsymbol{U}(N)$ for $N = 2^n$. The main results are achieved by showing that products of exponentiated sums of random permutations with random phases approximately match the first $2^{\Omega(n)}$ moments of the Haar measure. By substituting either $\tilde{O}(k)$-wise independent permutations or quantum-secure pseudorandom permutations (PRPs) in place of the random permutations, the authors obtain efficient constructions of unitary $k$-designs and parallel-secure PRUs. The key technical step involves demonstrating that an orthonormal basis for irreducible representations of the partition algebra has a low-degree large-$N$ expansion, allowing the distinguishing probability to be expressed as a low-degree rational polynomial of the dimension $N$. The paper also discusses the connection between the large-$N$ expansion in random matrix theory and the polynomial method, which is crucial for proving query lower bounds at finite-$N$.