We analyze the clustering of galaxies in the first public data release of the HSC Subaru Strategic Program. Despite the relatively small footprints of the observed fields, the data are an excellent proxy for the deep photometric datasets that will be acquired by LSST, and are therefore an ideal test bed for the analysis methods being implemented by the LSST DESC. We select a magnitude limited sample with $i<24.5$ and analyze it in four redshift bins covering $0.15lesssim z lesssim1.5$. We carry out a Fourier-space analysis of the two-point clustering of this sample, including all auto- and cross-correlations. We demonstrate the use of map-level deprojection methods to account for fluctuations in the galaxy number density caused by observational systematics. Through an HOD analysis, we place constraints on the characteristic halo masses of this sample, finding a good fit up to scales $k_{rm max}=1,{rm Mpc}^{-1}$, including both auto- and cross-correlations. Our results show monotonically decreasing average halo masses, which can be interpreted in terms of the drop-out of red galaxies at high redshifts for a flux-limited sample. In terms of photometric redshift systematics, we show that additional care is needed in order to marginalize over uncertainties in the redshift distribution in galaxy clustering, and that these uncertainties can be constrained by including cross-correlations. We are able to make a $sim3sigma$ detection of lensing magnification in the HSC data. Our results are stable to variations in $sigma_8$ and $Omega_c$ and we find constraints that agree well with measurements from Planck and low-redshift probes. Finally, we use our pipeline to study the clustering of galaxies as a function of limiting flux, and provide a simple fitting function for the linear galaxy bias for magnitude limited samples as a function of limiting magnitude and redshift. [abridged]
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