In recent years, high-resolution cosmic microwave background (CMB) measurements have opened up the possibility to explore statistical features of the temperature fluctuations down to very small angular scales. One method that has been used is the Wehrl entropy, which is, however, extremely costly in terms of computational time. Here, we propose several different pseudoentropy measures (projection, angular, and quadratic) that agree well with the Wehrl entropy, but are significantly faster to compute. All of the presented alternatives are rotationally invariant measures of entanglement after identifying each multipole of temperature fluctuations with a spin- quantum state and are very sensitive to non-Gaussianity, anisotropy, and statistical dependence of spherical harmonic coefficients in the data. We provide a simple proof that the projection pseudoentropy converges to the Wehrl entropy with increasing dimensionality of the ancilla projection space. Furthermore, for , we show that both the Wehrl entropy and the angular pseudoentropy can be expressed as one-dimensional functions of the squared chordal distance of multipole vectors, giving a tight connection between the two measures. We also show that the angular pseudoentropy can clearly distinguish between Gaussian and non-Gaussian temperature fluctuations at large multipoles and henceforth provides a non-brute-force method for identifying non-Gaussianities. This allows us to study possible hints of statistical anisotropy and non-Gaussianity in the CMB up to multipole using Planck 2015, Planck 2018, and WMAP 7-yr full sky data. We find that and have a large entropy at significance and a slight hint towards a connection of this with the cosmic dipole. On a wider range of large angular scales we do not find indications of violation of isotropy or Gaussianity. We also find a small-scale range, , that is incompatible with the assumptions at about level, although how much this significance can be reduced by taking into account the selection effect, i.e., how likely it is to find ranges of a certain size with the observed features, and inhomogeneous noise is left as an open question. Furthermore, we find overall similar results in our analysis of the 2015 and the 2018 data. Finally, we also demonstrate how a range of angular momenta can be studied with the range angular pseudoentropy, which measures averages and correlations of different multipoles. Our main purpose in this work is to introduce the methods, analyze their mathematical background, and demonstrate their usage for providing researchers in this field with an additional tool. We believe that the formalism developed here can underpin future studies of the Gaussianity and isotropy of the CMB and help to identify deviations, especially at small angular scales.