  Extracting meaningful information about unknown quantum states without
performing a full tomography is an important task. Low-dimensional projections
and random measurements can provide such insight but typically require careful
crafting. In this paper, we present an optical scheme based on sending unknown
input states through a multimode fiber and performing two-point intensity and
coincidence measurements. A short multimode fiber implements effectively a
random projection in the spatial domain, while a long-dispersive multimode
fiber performs a spatial and spectral projection. We experimentally show that
useful properties -- i.e., the purity, dimensionality, and degree of
indistinguishability -- of various states of light including spectrally
entangled biphoton states, can be obtained by measuring statistical properties
of photocurrents and their correlation between two outputs over many
realizations of unknown random projections. Moreover, we show that this
information can then be used for state classification.
