  In this paper, we shall prove two conjectures of Z.-W. Sun concerning
Ap\'ery-like series. One of the series is alternating whereas the other one is
not. Our main strategy is to convert the series (resp.~the alternating series)
to log-sine-cosine (resp.~log-sinh-cosh) integrals. Then we express all these
integrals in terms of single-valued Bloch-Wigner-Ramakrishnan-Wojtkowiak-Zagier
polylogarithms. The conjectures then follow from a few highly non-trivial
functional equations of the polylogarithms of weight $3$ and $4$.
