Disclosure: The following is just mostly a close retelling of the "Precise expression" section of the Wikipedia article on the microcanonical ensemble.
A macrostate is a probability distribution over microstates. It is not, intrinsically, specified by "temperature" or "volume", those are very special kinds of macrostates:
A classical macrostate is given by a phase space distribution $\rho(p,q)$ and often given by macroscopic variables (e.g. $E,V,N$ in the microcanonical ensemble approach), where you implicitly assume it's an equilibrium macrostate where every microstate that has these same macroscopic parameters within a range is equally likely, i.e. the probability distribution here is
$$ \rho(q,p) = \frac{1}{Z h^N C}f\left(H(q,p)-E,\omega\right),$$
for $h$ the coarse graining phase space volume, $C$ a coorection factor for indistinguishability, $Z$ a normalization factor and $\omega$ the range of energies we allow, where $f(x,\omega)$ is a function that peaks around $x=0$ with "width" $\omega$. Ideally, we'd take $\omega\to 0$, meaning $f$ is the Dirac delta.
In the same way, a quantum macrostate is a density matrix, since a probability distribution over pure states is what a density matrix is. For the microcanonical ensemble, one would take energy eigenstates $\lvert E_i\rangle$ and write
$$ \rho= \frac{1}{Z}\sum_i f(E_i - E,\omega) \lvert E_i\rangle\langle E_i\rvert,$$
but in this case the limit $\omega\to 0$ is somewhat dubious if the $E_i$ are discrete.
