  In this paper, we study the problem of user activity detection and
large-scale fading coefficient estimation in a random access wireless uplink
with a massive MIMO base station with a large number $M$ of antennas and a
large number of wireless single-antenna devices (users). We consider a block
fading channel model where the $M$-dimensional channel vector of each user
remains constant over a coherence block containing $L$ signal dimensions in
time-frequency. In the considered setting, the number of potential users
$K_\text{tot}$ is much larger than $L$ but at each time slot only
$K_a<<K_\text{tot}$ of them are active. Previous results, based on compressed
sensing, require that $K_a\leq L$, which is a bottleneck in massive deployment
scenarios such as Internet-of-Things and unsourced random access. In this work
we show that such limitation can be overcome when the number of base station
antennas $M$ is sufficiently large. We also provide two algorithms. One is
based on Non-Negative Least-Squares, for which the above scaling result can be
rigorously proved. The other consists of a low-complexity iterative
componentwise minimization of the likelihood function of the underlying
problem. Finally, we use the discussed approximated ML algorithm as the decoder
for the inner code in a concatenated coding scheme for unsourced random access,
a grant-free uncoordinated multiple access scheme where all users make use of
the same codebook, and the massive MIMO base station must come up with the list
of transmitted messages irrespectively of the identity of the transmitters. We
show that reliable communication is possible at any $E_b/N_0$ provided that a
sufficiently large number of base station antennas is used, and that a sum
spectral efficiency in the order of $\mathcal{O}(L\log(L))$ is achievable.
